Exam 2 Study Guide - MA 261 Spring 2026

Coverage: Lessons 17–33 (§15.8, §16.1–16.6, §17.1–17.6)

Exam Date: Tuesday, April 21, 2026

Time: 8:00 PM - 9:00 PM

Location: Announced on Brightspace

Lagrange Multipliers [Lesson 17] (§15.8)

Video Notes Practice

Method of Lagrange Multipliers

To find the extreme values of \(f(x,y)\) subject to a constraint \(g(x,y) = k\), solve the system:

\[\nabla f = \lambda\,\nabla g \qquad \text{and} \qquad g(x,y) = k\]

That is:

\(f_x = \lambda\, g_x\)
\(f_y = \lambda\, g_y\)
\(g(x,y) = k\)

Three Variables

To optimize \(f(x,y,z)\) subject to \(g(x,y,z) = k\):

\[\nabla f = \lambda\,\nabla g \qquad \text{and} \qquad g(x,y,z) = k\]

Note: Evaluate \(f\) at all solution points and compare to identify absolute max/min.

Practice Problems

Problem 1. Find the maximum value of \(f(x,y) = 3 + 2x - y\) subject to the constraint \(x^2 + 2xy + 4y^2 = 7\).

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Answer: 10

Problem 2. Find the maximum value of \(f(x,y) = 8x - 6y\) subject to the constraint \((x-1)^2 + y^2 = 1\).

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Answer: 18

Problem 3. The maximum and minimum values of \(f(x,y)=(x-2)^2+(y-4)^2\) subject to \(x^2+y^2=5\) are, respectively:

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Answer: Maximum 45, Minimum 5

Problem 4. Find the maximum of \(f(x,y,z) = x+y+z\) subject to \((x-1)^2+y^2+z^2=1\).

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Answer: \(1+\sqrt{3}\)

Problem 5. Find the absolute maximum value \(M\) and absolute minimum value \(m\) of \(f(x,y)=x+y\) subject to \(x^2-xy+y^2=1\).

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Answer: \(M=2,\;m=-2\)

Double Integrals over Rectangular Regions [Lesson 18] (§16.1)

Video Notes Practice

Double Integral over a Rectangle

For \(R = [a,b] \times [c,d]\):

\[\iint_R f(x,y)\,dA = \int_a^b \int_c^d f(x,y)\,dy\,dx = \int_c^d \int_a^b f(x,y)\,dx\,dy\]

Fubini's Theorem

If \(f\) is continuous on \(R = [a,b]\times[c,d]\), the order of integration may be switched:

\[\int_a^b\int_c^d f(x,y)\,dy\,dx = \int_c^d\int_a^b f(x,y)\,dx\,dy\]

Riemann Sum Definition

Partition \([a,b]\) into \(m\) subintervals and \([c,d]\) into \(n\) subintervals, each of width \(\Delta x = \frac{b-a}{m}\) and \(\Delta y = \frac{d-c}{n}\). Choose sample points \((x_{ij}^*, y_{ij}^*)\) in each subrectangle \(R_{ij}\):

\[\iint_R f(x,y)\,dA = \lim_{m,n\to\infty}\sum_{i=1}^{m}\sum_{j=1}^{n} f(x_{ij}^*,\, y_{ij}^*)\,\Delta A\]

where \(\Delta A = \Delta x\,\Delta y\) is the area of each subrectangle.

Using midpoints: \(x_i^* = \frac{x_{i-1}+x_i}{2}\), \(y_j^* = \frac{y_{j-1}+y_j}{2}\)
The double integral gives the signed volume between the surface \(z = f(x,y)\) and the \(xy\)-plane

Practice Problems

Problem 1. Compute \(\displaystyle\int_0^1\int_0^4 \sqrt{xy}\,dx\,dy\).

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Answer: \(\tfrac{32}{9}\)

Problem 2. Find the volume enclosed by \(x=0\), \(x=3\), \(y=0\), \(y=2\), \(z=1\), and \(z=e^{x+y}\).

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Answer: \(e^5-e^3-e^2-5\)

Problem 3. Find the average value of \(f(x,y)=x^2y\) over the rectangle \(R\) with vertices \((-1,0)\), \((-1,5)\), \((1,5)\), \((1,0)\).

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Answer: \(\tfrac{5}{6}\)

Problem 4. Evaluate \(\displaystyle\int_0^1\int_0^1\int_0^{2-y} e^x\,dz\,dx\,dy\).

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Answer: \(\tfrac{3}{2}(e-1)\)

Problem 5. Evaluate \(\displaystyle\iint_R \frac{x^2}{y^2+x^3}\,dA\) over \(R=\{1\le x\le 2,\;0\le y\le 4\}\).

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Answer: \(\tfrac{8}{3}(\ln10-\ln3)\)

Double Integrals over General Regions [Lesson 19] (§16.2)

Video Notes Practice

Type I Region (vertical strips)

Region bounded by functions of \(x\): \(g_1(x) \le y \le g_2(x)\), \(a \le x \le b\)

\[\iint_D f(x,y)\,dA = \int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx\]

Type II Region (horizontal strips)

Region bounded by functions of \(y\): \(h_1(y) \le x \le h_2(y)\), \(c \le y \le d\)

\[\iint_D f(x,y)\,dA = \int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy\]

Reversing Order of Integration

Sketch the region, identify bounds for the opposite order, then re-write the integral. This is often necessary when the inner integral has no elementary antiderivative.

Practice Problems

Problem 1. Evaluate (switching order of integration):

\(\displaystyle\int_0^{27}\int_{\sqrt[3]{x}}^{3} 4e^{y^4}\,dy\,dx\)

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Answer: \(e^{81}-1\)

Problem 2. Evaluate \(\displaystyle\iint_R \frac{\sin x}{x}\,dA\) where \(R\) is bounded by the \(x\)-axis, \(y=x\), and \(x=1\) (integrate in \(y\) first, then \(x\)).

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Answer: \(1-\cos(1)\)

Problem 3. Find the volume of the solid under \(z=2x\) and above the region in the first quadrant bounded by the coordinate axes and \(x+y=1\).

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Answer: \(\tfrac{1}{3}\)

Problem 4. Change the order of integration for \(\displaystyle\int_0^2\int_{x^2}^{2x}f(x,y)\,dy\,dx\).

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Answer: (original) — \(\displaystyle\int_0^4\int_{\sqrt{y}}^{y/2}f\,dx\,dy\)

Problem 5. Reverse the order and evaluate: \(\displaystyle\int_0^{2\sqrt{\ln3}}\!\int_{y/2}^{\sqrt{\ln3}} e^{x^2}\,dx\,dy\).

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Answer: \(2\)

Double Integrals in Polar Coordinates [Lesson 20] (§16.3)

Video Notes Practice

Polar Coordinate Substitution

Use \(x = r\cos\theta\), \(y = r\sin\theta\), \(x^2+y^2=r^2\):

\[\iint_D f(x,y)\,dA = \int_\alpha^\beta\int_{r_1(\theta)}^{r_2(\theta)} f(r\cos\theta,\, r\sin\theta)\,r\,dr\,d\theta\]

Key: Don't forget the Jacobian factor \(r\) in \(dA = r\,dr\,d\theta\).

When to Use Polar Coordinates

Region is a disk, annulus, or sector
Integrand contains \(x^2 + y^2\)
Region has circular symmetry

Practice Problems

Problem 1. Evaluate \(\displaystyle\int_0^3\!\int_0^{\sqrt{9-x^2}}\!\sqrt{x^2+y^2}\,dy\,dx\) by converting to polar.

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Answer: \(\tfrac{9\pi}{2}\)

Problem 2. Compute \(\displaystyle\iint_R e^{-(x^2+y^2)}\,dA\) where \(R\) is the disk \(x^2+y^2\le4\).

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Answer: \(\pi(1-e^{-4})\)

Problem 3. Rewrite in polar (do not evaluate): \(\displaystyle\int_0^1\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}(x^2+y^2)^{1/2}\,dx\,dy\).

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Answer: \(\displaystyle\int_0^\pi\!\int_0^1 r^2\,dr\,d\theta\)

Problem 4. Evaluate \(\displaystyle\int_0^{\sqrt{\pi}}\!\int_0^{\sqrt{\pi-y^2}}\sin(x^2+y^2)\,dx\,dy\).

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Answer: \(\tfrac{\pi}{2}\)

Problem 5. Compute \(\displaystyle\int_{-3}^{3}\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}} e^{x^2+y^2}\,dx\,dy\).

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Answer: \(\pi(e^9-1)\)

Triple Integrals [Lesson 21] (§16.4)

Video Notes Practice

Iterated Triple Integral

\[\iiint_E f(x,y,z)\,dV = \int_a^b\int_{g_1(x)}^{g_2(x)}\int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z)\,dz\,dy\,dx\]

Applications

Volume: \(V = \displaystyle\iiint_E dV\)
Average value: \(\bar{f} = \dfrac{1}{V}\displaystyle\iiint_E f\,dV\)

Note: There are 6 possible orders of integration — choose the order that fits the region and integrand most naturally.

Practice Problems

Problem 1. Evaluate \(\displaystyle\int_0^1\!\int_0^1\!\int_0^{2-y} e^x\,dz\,dx\,dy\).

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Answer: \(\tfrac{3}{2}(e-1)\)

Problem 2. Let \(E\) be the solid in the first octant above the \(xy\)-plane and below \(4x+2y+z=8\). For \(\iiint_E f\,dV=\int_0^a\!\int_0^b\!\int_0^c f\,dz\,dy\,dx\):

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Answer: \(a=2,\;b=4-2x\)

Problem 3. Let \(D\) be the solid bounded by \(x=0\), \(z=0\), \(y=x\), and \(x+y+z=2\). Write \(\iiint_D f\,dV\) as three different iterated integrals (in three different orders of integration).

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Answer: e.g. \(\displaystyle\int_0^1\!\int_x^{2-x}\!\int_0^{2-x-y}f\,dz\,dy\,dx\), \(\displaystyle\int_0^1\!\int_0^{2-2y}\!\int_0^{y}f\,dx\,dz\,dy\), \(\displaystyle\int_0^1\!\int_0^{2-x}\!\int_x^{2-x-z}f\,dy\,dz\,dx\)

Problem 4. Let \(V\) be bounded by \(z=6-x^2-y^2\) and \(z=x^2+y^2\). If \(V=\displaystyle\int_{-\sqrt{3}}^{\sqrt{3}}\!\int_a^b\!\int_{x^2+y^2}^{6-x^2-y^2}dz\,dy\,dx\), what are \(a\) and \(b\)?

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Answer: \(a=-\sqrt{3-x^2},\;b=\sqrt{3-x^2}\)

Problem 5. Evaluate \(\displaystyle\int_{-1}^{1}\!\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\!\int_{-1}^{1}(x^2+y^2)^{3/2}\,dz\,dx\,dy\).

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Answer: \(\tfrac{4\pi}{5}\)

Triple Integrals in Cylindrical Coordinates [Lesson 22] (§16.5)

Video Notes Practice

Cylindrical Coordinates

Relation to rectangular coordinates:

\[x = r\cos\theta,\quad y = r\sin\theta,\quad z = z\] \[r^2 = x^2+y^2,\quad \tan\theta = \frac{y}{x}\]

Triple Integral in Cylindrical Coordinates

\[\iiint_E f(x,y,z)\,dV = \int_\alpha^\beta\int_{r_1}^{r_2}\int_{u_1}^{u_2} f(r\cos\theta,r\sin\theta,z)\,r\,dz\,dr\,d\theta\]

Key: Don't forget the Jacobian factor \(r\) in \(dV = r\,dz\,dr\,d\theta\).

When to Use Cylindrical Coordinates

Region has an axis of symmetry along the \(z\)-axis
Integrand or region involves \(x^2 + y^2\)
Region is bounded by a cylinder \(x^2+y^2 = a^2\), cone, or paraboloid

Practice Problems

Problem 1. Rewrite \(\displaystyle\int_0^{\sqrt{2}}\!\int_0^{\sqrt{4-x^2}}\!\int_{2}^{6}\!e^{-x^2-y^2}\,dz\,dy\,dx\) in cylindrical coordinates.

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Answer: \(\displaystyle\int_2^6\!\int_{\pi/4}^{\pi/2}\!\int_0^2 re^{-r^2}\,dr\,d\theta\,dz\)

Problem 2. Evaluate \(\displaystyle\int_0^2\!\int_0^{\sqrt{4-x^2}}\!\int_{\sqrt{x^2+y^2}}^{\sqrt{2(x^2+y^2)}} dz\,dy\,dx\) using cylindrical coordinates.

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Answer: \(\tfrac{4\pi}{3}(\sqrt{2}-1)\)

Problem 3. Convert to cylindrical and evaluate \(\displaystyle\int_0^{10}\!\int_0^{\sqrt{100-x^2}}\!\int_0^{\sqrt{x^2+y^2}}\!\frac{1}{\sqrt{x^2+y^2}}\,dz\,dy\,dx\).

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Answer: \(25\pi\)

Problem 4. Find the volume of the solid bounded by \(z=x^2+y^2-9\) and \(z=-2(x^2+y^2)\) using cylindrical coordinates.

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Answer: \(\tfrac{81\pi}{2}\)

Problem 5. Write the volume of the solid bounded by the cone \(z^2=x^2+y^2\) between the planes \(z=1\) and \(z=2\) as a spherical-coordinates integral.

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Answer: \(\displaystyle\int_0^{2\pi}\!\int_0^{\pi/4}\!\int_{\sec\phi}^{2\sec\phi}\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\)

Triple Integrals in Spherical Coordinates [Lesson 23] (§16.5)

Video Notes Practice

Spherical Coordinates

\[x = \rho\sin\phi\cos\theta,\quad y = \rho\sin\phi\sin\theta,\quad z = \rho\cos\phi\] \[\rho^2 = x^2+y^2+z^2, \quad \rho \ge 0,\quad 0 \le \phi \le \pi,\quad 0 \le \theta \le 2\pi\]

Triple Integral in Spherical Coordinates

\[\iiint_E f\,dV = \int_0^{2\pi}\int_0^{\phi_{\max}}\int_0^{\rho_{\max}} f\,\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\]

Key: The Jacobian is \(\rho^2\sin\phi\). Don't forget it!

Useful Identities

Sphere: \(\rho = a\)
Cone: \(\phi = \phi_0\) (half-angle from positive \(z\)-axis)
\(z = \rho\cos\phi\), \(\;r = \rho\sin\phi\)

Practice Problems

Problem 1. The volume of the region inside \(x^2+y^2+z^2=4\) and inside the cone \(z=\sqrt{3x^2+3y^2}\) is \(\displaystyle\int_0^{2\pi}\!\int_0^a\!\int_0^b c\,d\rho\,d\phi\,d\theta\). What are \(a\), \(b\), and \(c\)?

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Answer: \(a=\pi/6,\;b=2,\;c=\rho^2\sin\phi\)

Problem 2. Write the volume of the solid bounded by the cone \(z=\sqrt{x^2+y^2}\) and the sphere \(x^2+y^2+z^2=8\) as a spherical-coordinates integral.

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Answer: \(\displaystyle\int_0^{2\pi}\!\int_0^{\pi/4}\!\int_0^{\sqrt{8}}\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\)

Problem 3. Write the volume of the region inside the sphere of radius 2 centered at \((0,0,2)\) and outside the sphere of radius 2 centered at the origin as a spherical-coordinates integral.

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Answer: \(\displaystyle\int_0^{2\pi}\!\int_0^{\pi/3}\!\int_2^{4\cos\phi}\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\)

Problem 4. Converting \(\displaystyle\int_{-1}^{1}\!\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\!\int_{\sqrt{3(x^2+y^2)}}^{\sqrt{4-x^2-y^2}} f\,dz\,dy\,dx\) to spherical gives \(\displaystyle\int_0^a\!\int_0^b\!\int_0^c f\cdot\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\) (order: \(d\rho\,d\phi\,d\theta\), so \(a\) is the upper limit on \(\theta\), \(b\) on \(\phi\), \(c\) on \(\rho\)). Find \(\dfrac{bc}{a}\).

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Answer: \(\tfrac{1}{6}\) (\(a=2\pi,\;b=\tfrac{\pi}{6},\;c=2\))

Problem 5. Find the volume of the solid region enclosed by the surface \(\rho=12\cos\phi\).

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Answer: \(288\pi\)

Integrals in Mass Calculations [Lesson 24] (§16.6)

Video Notes Practice

Mass, Moments, and Center of Mass

For a lamina (2D) with density \(\delta(x,y)\):

\[m = \iint_D \delta(x,y)\,dA\] \[\bar{x} = \frac{1}{m}\iint_D x\,\delta(x,y)\,dA, \qquad \bar{y} = \frac{1}{m}\iint_D y\,\delta(x,y)\,dA\]

3D Solid

For a solid \(E\) with density \(\delta(x,y,z)\):

\[m = \iiint_E \delta(x,y,z)\,dV\] \[\bar{x} = \frac{1}{m}\iiint_E x\,\delta\,dV,\quad \bar{y} = \frac{1}{m}\iiint_E y\,\delta\,dV,\quad \bar{z} = \frac{1}{m}\iiint_E z\,\delta\,dV\]

Moments of Inertia

\[I_x = \iint_D y^2\,\delta\,dA,\quad I_y = \iint_D x^2\,\delta\,dA,\quad I_0 = \iint_D (x^2+y^2)\,\delta\,dA\]

Practice Problems

Problem 1. Find the center of mass of the rectangle \(\{(x,y)\mid 0\le x\le 2,\;0\le y\le 4\}\) with density \(\rho(x,y)=1+y\).

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Answer: \(\left(1,\;\tfrac{22}{9}\right)\)

Problem 2. A cube \(0\le x,y,z\le 1\) has density \(\delta=x+yz\). Find the \(x\)-coordinate of the center of mass, given \(\iiint_{\text{cube}}\delta\,dV=\tfrac{3}{4}\).

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Answer: \(\tfrac{11}{18}\)

Problem 3. Find the \(x\)-coordinate of the center of mass of the plate bounded by \(y=x\), \(y=\tfrac{1}{2}x\), \(x=1\) with density \(\delta(x,y)=2x\). (Given: \(M_x=\iint y\delta\,dA\), \(M_y=\iint x\delta\,dA\).)

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Answer: \(\tfrac{3}{4}\)

Problem 4. Calculate the mass of the tetrahedron with corners \((0,0,0)\), \((1,0,0)\), \((0,2,0)\), \((0,0,4)\) with density \(\rho(x,y,z)=2z\).

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Answer: \(\tfrac{8}{3}\)

Problem 5. A lamina with density \(\rho(x,y)=xy\) occupies the region bounded by \(y=x^2\), \(y=1\), \(x=0\). The mass is \(\tfrac{1}{6}\). Find the \(y\)-coordinate of the center of mass.

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Answer: \(\tfrac{3}{4}\)

Vector Fields [Lesson 25] (§17.1)

Video Notes Practice

Definition

A vector field assigns a vector to each point in a region:

\[\vec{F}(x,y) = P(x,y)\,\vec{i} + Q(x,y)\,\vec{j} = \langle P, Q \rangle\] \[\vec{F}(x,y,z) = P\,\vec{i} + Q\,\vec{j} + R\,\vec{k} = \langle P, Q, R \rangle\]

Conservative Vector Fields

A vector field \(\vec{F}\) is conservative if there exists a scalar function \(f\) (the potential function) such that:

\[\vec{F} = \nabla f \qquad \Longleftrightarrow \qquad P = f_x,\; Q = f_y \; (\text{and }R = f_z)\]

Test for Conservative Field in 2D

\[\vec{F} = \langle P, Q \rangle \text{ is conservative} \iff \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}\]

Practice Problems

Problem 1. Consider \(\vec{F}=\langle e^{yz}-y\sin(xy),\;zxe^{yz}-x\sin(xy),\;xye^{yz}\rangle\). Find a potential function \(\varphi\) if one exists.

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Answer: \(xe^{yz}+\cos(xy)\)

Problem 2. Let \(a\ne0\) and \(C\) be the curve \(x^2-2y^2=a\). Which \(\vec{F}\) is orthogonal to the tangent of \(C\) at every point \((x_0,y_0)\) on \(C\)?

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Answer: \(\langle 2x,-4y\rangle\)

Problem 3. Which vector field corresponds to the field with vectors rotating counterclockwise (pointing left above the \(x\)-axis, right below it)?

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Answer: \(\langle -y,x\rangle\)

Problem 4. Which vector field corresponds to the pictured field (vectors point in the \(x\)-direction with magnitude growing in \(y\))?

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Answer: \(\langle y,x\rangle\)

Problem 5. Suppose \(f(x,y,z)=xy^2z^3\). Find \(\operatorname{div}(\nabla f)\) at the point \((2,-1,1)\).

▼ Show answer

Answer: 16

Line Integrals of Functions [Lesson 26] (§17.2)

Video Notes Practice

Line Integral of a Scalar Function

For curve \(C\) parametrized by \(\vec{r}(t)\), \(a \le t \le b\):

\[\int_C f(x,y,z)\,ds = \int_a^b f(\vec{r}(t))\,|\vec{r}'(t)|\,dt\]

Interpretations

Arc length: \(\displaystyle\int_C ds = \int_a^b |\vec{r}'(t)|\,dt\)
Mass of a wire with linear density \(\delta\): \(m = \displaystyle\int_C \delta\,ds\)

Practice Problems

Problem 1. Compute \(\displaystyle\int_C f(x,y)\,ds\) where \(f(x,y)=\sin(2\pi x)+xy\) and \(C\colon\vec{r}(t)=\langle t,t\rangle\), \(0\le t\le1\).

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Answer: \(\tfrac{\sqrt{2}}{3}\)

Problem 2. Let \(C\) be the helix \(\vec{r}(t)=\langle\cos t,\sin t,t\rangle\), \(0\le t\le\pi/2\). Compute \(\displaystyle\int_C xy\,ds\).

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Answer: \(\tfrac{\sqrt{2}}{2}\)

Problem 3. Calculate \(\displaystyle\int_C\frac{1}{(x-y)^2}\,ds\) where \(C\) is the straight line segment from \((1,0)\) to \((5,3)\).

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Answer: \(\tfrac{5}{2}\)

Problem 4. Let \(C\) be the half-circle \(x^2+y^2=4\) with \(x\ge0\). Compute \(\displaystyle\int_C x\,ds\).

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Answer: \(8\)

Problem 5. Find the arc length of \(\vec{r}(t) = \langle 4\cos t,\, 4\sin t,\, 3t\rangle\) for \(0\le t\le 2\pi\).

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Answer: \(10\pi\)

Line Integrals of Vector Fields [Lesson 27] (§17.2)

Video Notes Practice

Line Integral of a Vector Field (Work)

The work done by \(\vec{F}\) along curve \(C\):

\[\int_C \vec{F}\cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t))\cdot\vec{r}'(t)\,dt = \int_C P\,dx + Q\,dy + R\,dz\]

Key Properties

Reversing orientation changes the sign: \(\displaystyle\int_{-C}\vec{F}\cdot d\vec{r} = -\int_C \vec{F}\cdot d\vec{r}\)
Linearity: \(\displaystyle\int_C (a\vec{F}+b\vec{G})\cdot d\vec{r} = a\int_C\vec{F}\cdot d\vec{r} + b\int_C\vec{G}\cdot d\vec{r}\)

Practice Problems

Problem 1. Given force field \(\vec{F}(x,y)=\langle-y,x\rangle\), find the work to move an object along the ellipse \(\vec{r}(t)=\langle2\cos t,3\sin t\rangle\) from \((2,0)\) to \((0,3)\).

▼ Show answer

Answer: \(3\pi\)

Problem 2. Compute the flux of \(\vec{F}(x,y)=\langle x,y\rangle\) across the curve \(\vec{r}(t)=\langle3\cos t,2\sin t\rangle\), \(-\pi\le t\le\pi\). (Flux \(=\displaystyle\int_C P\,dy-Q\,dx\).)

▼ Show answer

Answer: \(12\pi\)

Problem 3. Given force field \(\vec{F}(x,y,z)=\langle y,z,x\rangle\), find the work to move an object along the straight line from \((0,0,0)\) to \((2,3,4)\).

▼ Show answer

Answer: \(\tfrac{29}{2}\)

Problem 4. Find \(\displaystyle\int_C\vec{F}\cdot\vec{T}\,ds\) where \(\vec{F}(x,y,z)=\langle ye^z,e^y+xe^z,xye^z\rangle\) on a smooth curve from \((0,0,0)\) to \((-1,1,1)\).

▼ Show answer

Answer: \(-e\)

Problem 5. Evaluate \(\displaystyle\int_C\vec{F}\cdot d\vec{r}\) where \(\vec{F}(x,y,z)=\langle2xy-yz,\;x^2-xz,\;-xy\rangle\) on a smooth curve from \((2,1,0)\) to \((3,2,-1)\).

▼ Show answer

Answer: 20

Fundamental Theorem of Line Integrals [Lesson 28] (§17.3)

Video Notes Practice

Fundamental Theorem of Line Integrals

If \(\vec{F} = \nabla f\) (conservative) and \(C\) goes from \(A\) to \(B\):

\[\int_C \nabla f \cdot d\vec{r} = f(B) - f(A)\]

Independence of Path

\(\displaystyle\int_C \vec{F}\cdot d\vec{r}\) is path-independent if and only if \(\vec{F}\) is conservative.

Closed Curves

\[\oint_C \vec{F}\cdot d\vec{r} = 0 \iff \vec{F} \text{ is conservative}\]

Finding the Potential Function

Given \(\vec{F} = \langle P, Q \rangle\) is conservative:

Integrate \(P\) with respect to \(x\) to get \(f(x,y) = \int P\,dx + g(y)\)
Differentiate with respect to \(y\) and set equal to \(Q\) to find \(g'(y)\), then integrate

Practice Problems

Problem 1. Find \(\displaystyle\int_C\vec{F}\cdot d\vec{r}\) where \(\vec{F}(x,y,z)=\langle2xy-yz,\;x^2-xz,\;-xy\rangle\) on a smooth curve from \((2,1,0)\) to \((3,2,-1)\).

▼ Show answer

Answer: 20

Problem 2. For \(\vec{F}=\langle e^{yz}-y\sin(xy),\;zxe^{yz}-x\sin(xy),\;xye^{yz}\rangle\), find a potential function.

▼ Show answer

Answer: \(\varphi=xe^{yz}+\cos(xy)\)

Problem 3. A potential for \(\vec{F}=\langle\sin y,\;x\cos y\rangle\) is:

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Answer: \(x\sin y+1\)

Problem 4. Find \(\displaystyle\int_C\vec{F}\cdot\vec{T}\,ds\) where \(\vec{F}=\langle ye^z,\;e^y+xe^z,\;xye^z\rangle\) on a smooth curve from \((0,0,0)\) to \((-1,1,1)\).

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Answer: \(-e\)

Problem 5. Evaluate \(\displaystyle\int_C\nabla(x^2y-xyz)\cdot d\vec{r}\) where \(C\) is any path from \((1,0,0)\) to \((2,1,-1)\).

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Answer: 6

Green's Theorem [Lesson 29] (§17.4)

Video Notes Practice

Green's Theorem

Let \(C\) be a positively oriented (counterclockwise), simple closed curve bounding region \(D\):

\[\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA\]

Area via Green's Theorem

\[A = \oint_C x\,dy = -\oint_C y\,dx = \frac{1}{2}\oint_C x\,dy - y\,dx\]

Note: Green's theorem converts a difficult line integral into a double integral, or vice versa.

Practice Problems

Problem 1. Use Green's Theorem to evaluate \(\displaystyle\oint_C x\,dx+(x^2+y^2)\,dy\) where \(C\) is the boundary of the rectangle with vertices \((0,0)\), \((2,0)\), \((2,3)\), \((0,3)\), counterclockwise.

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Answer: 24

Problem 2. Let \(R\) be the triangle with vertices \((0,0)\), \((2,0)\), \((1,1)\) and \(C\) its CCW boundary. Compute \(\displaystyle\int_C\cos^3(x)\,dx+e^x\,dy\).

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Answer: \(e^2-2e+1\)

Problem 3. Use Green's Theorem to evaluate \(\displaystyle\oint_C y\,dx + 2x\,dy\) where \(C\) is the square with vertices \((\pm1,\pm1)\), counterclockwise.

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Answer: \(4\)

Problem 4. Use Green's theorem to evaluate \(\displaystyle\oint_C(y^3)\,dx-(x^3)\,dy\) where \(C\) is the circle \(x^2+y^2=1\), CCW.

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Answer: \(-3\pi\)

Problem 5. Use Green's theorem to find the area of the ellipse \(\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1\).

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Answer: \(\pi ab\)

Curl and Divergence [Lesson 30] (§17.5)

Video Notes Practice

Curl

For \(\vec{F} = \langle P, Q, R \rangle\):

\[\text{curl}\,\vec{F} = \nabla \times \vec{F} = \begin{vmatrix}\vec{i} & \vec{j} & \vec{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R\end{vmatrix} = \left\langle R_y - Q_z,\, P_z - R_x,\, Q_x - P_y \right\rangle\]

Divergence

\[\text{div}\,\vec{F} = \nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]

Key Identities

\(\vec{F}\) conservative \(\Rightarrow\) \(\text{curl}\,\vec{F} = \vec{0}\)
\(\text{div}(\text{curl}\,\vec{F}) = 0\) always
\(\text{curl}(\nabla f) = \vec{0}\) always (gradient fields are irrotational)

Practice Problems

Problem 1. If \(\vec{F}=\langle xy,\sin z,\cos y\rangle\), find \(\operatorname{curl}(\vec{F})\).

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Answer: \(\langle-\sin y-\cos z,\,0,\,-x\rangle\)

Problem 2. Let \(f(x,y,z)=xy^2z^3\). Compute \(\operatorname{div}(\nabla f)\) at \((2,-1,1)\).

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Answer: 16

Problem 3. For \(\vec{F}=\langle e^{yz}-y\sin(xy),\;zxe^{yz}-x\sin(xy),\;xye^{yz}\rangle\), verify that \(\operatorname{curl}\vec{F}=\vec{0}\) and state what this implies.

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Answer: \(\vec{F}\) is conservative

Problem 4. Determine if \(\vec{F}=\langle yz,xz,xy\rangle\) is conservative using \(\operatorname{curl}\vec{F}\). If so, find \(\varphi\).

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Answer: Conservative; \(\varphi=xyz+C\)

Problem 5. Find \(\operatorname{div}\vec{F}\) for \(\vec{F}=\langle e^{xy},e^{yz},e^{xz}\rangle\), and verify the identity \(\operatorname{div}(\operatorname{curl}\vec{G})=0\) for any smooth \(\vec{G}\).

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Answer: \(\operatorname{div}\vec{F}=ye^{xy}+ze^{yz}+xe^{xz}\)

Surface Integrals [Lessons 31–33] (§17.6)

Video (Lesson 31) Notes (Lesson 31) Video (Lesson 32) Notes (Lesson 32) Video (Lesson 33) Notes (Lesson 33) Practice

Surface Integral of a Scalar Function

For surface \(z = g(x,y)\) over region \(D\):

\[\iint_S f(x,y,z)\,dS = \iint_D f(x,y,g(x,y))\sqrt{g_x^2 + g_y^2 + 1}\,dA\]

Surface Area

\[A(S) = \iint_S dS = \iint_D \sqrt{g_x^2 + g_y^2 + 1}\,dA\]

Flux Integral (Surface Integral of a Vector Field)

For \(\vec{F} = \langle P, Q, R \rangle\) and oriented surface \(S\) with unit normal \(\hat{n}\):

\[\iint_S \vec{F}\cdot d\vec{S} = \iint_S \vec{F}\cdot\hat{n}\,dS\]

For \(z = g(x,y)\) with upward orientation (\(\hat{n}\) pointing upward):

\[\iint_S \vec{F}\cdot d\vec{S} = \iint_D \left(-P g_x - Q g_y + R\right)dA\]

Parametric Surface

For surface \(\vec{r}(u,v)\):

\[\iint_S f\,dS = \iint_D f(\vec{r}(u,v))\,|\vec{r}_u \times \vec{r}_v|\,dA\]

\[\iint_S \vec{F}\cdot d\vec{S} = \iint_D \vec{F}(\vec{r}(u,v))\cdot(\vec{r}_u \times \vec{r}_v)\,dA\]

Practice Problems

Problem 1. Find the area of the part of the plane \(2x+5y+z=10\) that lies in the first octant.

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Answer: \(5\sqrt{30}\)

Problem 2. Compute the flux of \(\vec{F}(x,y)=\langle x,y\rangle\) across \(\vec{r}(t)=\langle3\cos t,2\sin t\rangle\), \(-\pi\le t\le\pi\). (Flux \(=\displaystyle\int_C P\,dy-Q\,dx\).)

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Answer: \(12\pi\)

Problem 3. Evaluate \(\iint_S z\,dS\) where \(S\) is the portion of the paraboloid \(z=x^2+y^2\) below \(z=4\).

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Answer: \(\dfrac{\pi}{6}(17\sqrt{17}-1)\)

Problem 4. Find the flux of \(\vec{F}=\langle x,y,z\rangle\) upward through the surface \(z=4-x^2-y^2\) above the \(xy\)-plane.

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Answer: \(\dfrac{40\pi}{3}\) (approx — verify)

Problem 5. Find the surface area of the cone \(z=\sqrt{x^2+y^2}\) below \(z=3\).

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Answer: \(9\pi\sqrt{2}\)

Problem 6. Evaluate \(\iint_S\vec{F}\cdot d\vec{S}\) where \(\vec{F}=\langle y,-x,1\rangle\) and \(S\) is the disk \(z=1\), \(x^2+y^2\le1\), oriented upward.

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Answer: \(\pi\)