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Lesson 21: Triple integrals

Warmup: What is the area between \(y = x^2\) and \(y = 4\).

A diagram on a grid background depicts a shaded region \(D\) in a 2D Cartesian coordinate system. The region is bounded below by the parabola \(y = x^2\) and above by the horizontal line \(y = 4\). The parabola opens upward with its vertex at the origin \((0,0)\). Vertical dashed blue lines indicate the boundaries along the x-axis at \(x = -2\) and \(x = 2\). A vertical rectangular strip is drawn within the shaded region to represent a differential element of area; arrows point from the top of this strip to the line \(y = 4\) and from the bottom of the strip to the curve \(y = x^2\).
\[ = \int_{-2}^{2} (4 - x^2) \, dx \]

Observe! \[ 4 - x^2 = \int_{x^2}^{4} 1 \, dy \]

\[ = \int_{-2}^{2} \int_{x^2}^{4} 1 \, dy \, dx \] \[ = \iint_{D} 1 \, dA \]

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What is the volume of the Region between \( z = x^2 + y^2 \) & \( z = 4 \)?

Visual Description:

The page features two diagrams illustrating a multivariable calculus problem:

  • 3D Coordinate System: Shows a paraboloid defined by the equation \( z = x^2 + y^2 \). The paraboloid opens upwards from the origin \((0,0,0)\). A horizontal plane, shown in blue and labeled \( z = 4 \), caps the paraboloid. The region enclosed between the paraboloid and the plane is labeled \( R \). A purple horizontal cross-sectional disk is drawn at an arbitrary height \( z \), labeled \( A(z) \), with a small vertical thickness indicated as \( \Delta z \).
  • 2D Coordinate System: Displays the projection of the cross-section onto the \( xy \)-plane. This projection is a shaded purple disk centered at the origin, labeled \( D \).

Example Solution:

To find the volume of the region \( R \), we set up the integral based on horizontal cross-sections:

\[ \text{Volume} = \int_{0}^{4} A(z) \, dz \]

In this expression, \( A(z) \) represents the Area of cross section, which can be defined as a Double integral:

\[ = \int_{0}^{4} \iint_{D} 1 \, dz \]

= Triple integral.

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Volume and Mass of a 3d Domain D

\[ \text{Volume} = \iiint_{D} 1 \, dV \]

Suppose \(f(x, y, z)\) is the density of the domain

\[ \text{Mass} = \iiint_{D} f(x, y, z) \, dV \]

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Triple integral as Riemann Sum:

* \(f(x)\) on \([a, b]\)

\[ \int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x \]
Visual Description: A horizontal blue line segment represents the interval from point \(a\) on the left to point \(b\) on the right. A small section of the line is marked with a bracket. Accompanying blue text notes:
  • * Divide into small intervals of length \(\Delta x\)
  • * \(f(x_i)\) height at sample point

* \(f(x, y)\) on \([a, b] \times [c, d]\)

Divide into small Rectangle of Area \(\Delta x \Delta y\)

\[ \iint_{R} f(x, y) dA = \lim_{m \to \infty} \lim_{n \to \infty} \sum_{j=1}^{m} \sum_{i=1}^{n} f(x_i, y_j) \Delta x \Delta y \]
Visual Description: A large grey rectangle represents the region \(R\). Within the rectangle, a single small sub-rectangle is colored red and labeled with its dimensions: width \(\Delta x\) and height \(\Delta y\).

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Triple Integrals over Rectangular Boxes

\[D = [a, b] \times [c, d] \times [e, f]\] \[\iiint_{D} f(x, y, z) \, dV\]
A three-dimensional rectangular prism represents the domain \(D\) drawn on a grid. The prism is partitioned into sub-boxes, with specific dimensions labeled:
  • The depth increment is labeled \(\Delta x\) in blue.
  • The width increment is labeled \(\Delta y\) in green.
  • The height increment is labeled \(\Delta z\) in purple.
  • A small representative sub-box in the interior of the prism is highlighted with a red outline.
Divide the domain to small Boxes of volume \[\Delta V = \Delta x \Delta y \Delta z\]
\[\lim_{l \to \infty} \lim_{m \to \infty} \lim_{n \to \infty} \sum_{k=1}^{l} \sum_{j=1}^{n} \sum_{i=1}^{m} f(x_i, y_j, z_k) \Delta x \Delta y \Delta z\]

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Triple integrals as iterated integrals

Example: \( B = [0,2] \times [0,1] \times [0,3] \), evaluate \( \iiint_B xyz^2 \, dV \)
A 3D diagram shows a rectangular box in a right-handed Cartesian coordinate system. The vertical \( z \)-axis is labeled with a point at \( (0,0,3) \). The \( x \)-axis extends forward and to the left, labeled with a point at \( (2,0,0) \). The \( y \)-axis extends to the right, and the far corner of the box's base is labeled \( (2,1,0) \). A vertical purple line segment with endpoints marked by dots spans the height of the box, illustrating integration with respect to \( z \). The base of the box in the \( xy \)-plane is shaded with red diagonal hatch marks to represent the "shadow" of the region.
Fix \( x,y \to \) integrate w.r.t \( z \) first then double integral on the shadow in \( xy \) plane

\( 0 \le z \le 3 \)

shadow:
A 2D coordinate graph shows the "shadow" region \( R \) in the \( xy \)-plane. The region \( R \) is a rectangle shaded in light red, bounded by \( x=0 \) to \( x=2 \) on the horizontal axis and \( y=0 \) to \( y=1 \) on the vertical axis.
\[ \iiint_B xyz^2 \, dV = \iint_R \int_0^3 xyz^2 \, dz \, dA = \iint_R 9xy \, dA \]

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Shadows

An xy-coordinate plane graph displays a rectangular region labeled R. The region R is shaded in red and is bounded by the x-axis from 0 to 2 and the y-axis from 0 to 1. The horizontal axis is labeled x with a tick mark at 2, and the vertical axis is labeled y with a tick mark at 1.
\[ \iint_{R} \int_{0}^{3} xyz^{2} \, dz \, dA = \iint_{R} 9xy \, dA \]

Fix \(x\), integrate w.r.t \(y\) first on \(0 \le y \le 1\) then w.r.t \(x\) on \(0 \le x \le 2\)

\[ = \int_{0}^{2} \int_{0}^{1} 9xy \, dy \, dx \] \[ = \int_{0}^{2} \frac{9x}{2} \, dx = 9 \]

Fix \(y\), integrate w.r.t \(x\) first then w.r.t \(y\)

\(0 \le x \le 2\)

\(0 \le y \le 1\)

\[ \int_{0}^{1} \int_{0}^{2} 9xy \, dx \, dy \] \[ = \int_{0}^{1} 18y \, dy \] \[ = 9 \]

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Integration Order: Integrating with Respect to \(y\) First

Fix \(x\) and \(z\) first instead and integrate w.r.t. \(y\) and then on shadow in \(xz\) plane.

3D Coordinate Plot: A rectangular prism is depicted in a three-dimensional \(xyz\) coordinate system. The vertices of the prism are shown with labels at \((0,0,3)\), \((2,0,0)\), and \((2,1,0)\). The face of the prism lying on the \(xz\)-plane is marked with red scribbled lines, and a green arrow points through the volume in the positive \(y\) direction, indicating the direction of integration. This highlighted face represents the "shadow" of the volume onto the \(xz\)-plane.

Shadow Plot (xz-plane): A 2D graph titled "Shadow:" illustrates the region \(R\) in the \(xz\)-plane. The region is a rectangle bounded by \(x=0\) to \(x=2\) on the horizontal axis and \(z=0\) to \(z=3\) on the vertical axis. The area is shaded light red and labeled \(R\).

Bounds for \(y\):

\[0 \le y \le 1\]

The triple integral is set up by first integrating with respect to \(y\) over the interval \([0, 1]\), and then over the rectangular region \(R\) in the \(xz\)-plane:

\[ \iint_{R} \int_{0}^{1} xyz^2 \, dy \, dA \]

Substituting the bounds for the region \(R\) (\(0 \le x \le 2\) and \(0 \le z \le 3\)), the iterated integral can be written in two ways:

\[ = \int_{0}^{2} \int_{0}^{3} \int_{0}^{1} xyz^2 \, dy \, dz \, dx \] \[ = \int_{0}^{3} \int_{0}^{2} \int_{0}^{1} xyz^2 \, dy \, dx \, dz \]

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Triple Integrals: Integration Order

fix \(y\) & \(z\) first and integrate w.r.t. \(x\) and then on shadow in \(yz\) plane
3D Coordinate Plot: A three-dimensional Cartesian coordinate system displays a rectangular prism. The axes are labeled \(x\), \(y\), and \(z\). The prism's base lies on the \(xy\)-plane with vertices labeled at \((2,0,0)\) and \((2,1,0)\). The face in the \(yz\)-plane is shaded gray. There is a red hatched region drawn on the front face of the box. A blue line segment with endpoints marked by blue dots spans from the \(z\)-axis diagonally across the volume toward the front-right edge.
Shadow Region Diagram: A 2D plot titled "Shadow:" illustrates the projection of the volume onto the \(yz\)-plane. A rectangular region \(R\) is shaded light red in the first quadrant. The vertical axis is labeled \(z\) with a tick mark at 3, and the horizontal axis is labeled \(y\) with a tick mark at 2.

\(0 \le x \le 2\)

\[ \iint\limits_{R} \int_{0}^{2} xy z^{2} \, dx \, dA \] \[ = \int_{0}^{3} \int_{0}^{1} \int_{0}^{2} xy z^{2} \, dx \, dy \, dz \] \[ = \int_{0}^{1} \int_{0}^{3} \int_{0}^{2} xy z^{2} \, dx \, dz \, dy \]

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Example: Find volume of solid under \(x+y+z=1\) in 1st octant.

A 3D coordinate system graph illustrates a solid tetrahedron in the first octant bounded by the plane \(x+y+z=1\). The plane intersects the coordinate axes at points \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). The region on the \(xy\)-plane directly beneath the plane is shaded with red diagonal lines and labeled as the "shadow". A representative horizontal cross-section is shown in purple at a height \(z\), and the diagram indicates the bounds of integration from the \(xy\)-plane up to the boundary plane.
\[ \text{Volume} = \iiint_D 1 \, dV \]

fix \(x,y\) so integrate w.r.t \(z\) first

\(0 \le z \le \text{hitting plane } x+y+z=1\)
\(0 \le z \le 1-x-y\)

then Double integral on shadow in \(xy\)

A 2D graph titled "shadow" shows the projection of the solid onto the \(xy\)-plane, labeled region \(D\). The region is a triangle with vertices at the origin \((0,0)\), \((1,0)\) on the \(x\)-axis, and \((0,1)\) on the \(y\)-axis. The boundary line is \(y = 1 - x\). A vertical line segment is drawn at a general position \(x\) from the \(x\)-axis to the boundary line, representing the inner integration limits for \(y\).
\[ \iint_D \int_0^{1-x-y} 1 \, dz \, dA = \int_0^1 \int_0^{1-x} \int_0^{1-x-y} 1 \, dz \, dy \, dx = \int_0^1 \int_0^{1-x} 1-x-y \, dy \, dx \] \[ = \int_0^1 (1-x)(1-x) - \frac{(1-x)^2}{2} \, dx \] \[ = \int_0^1 \frac{(1-x)^2}{2} \, dx = \frac{1}{6} \]

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Another Way!

Visual Description: A 3D coordinate system showing a tetrahedron in the first octant. The vertices of the tetrahedron are located at \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). The slanted face is part of the plane defined by the equation \(x + y + z = 1\). A horizontal, purple-shaded cross-section is shown at an arbitrary height \(z\), labeled as \(A(z)\).
\[ \text{Volume} = \iiint_D 1 \, dV \] \[ = \int_0^1 A(z) \, dz \]
compute this using double integrals.
Cross-Section: \(0 \le z \le 1\)
Visual Description: A 2D \(xy\)-plane diagram representing the cross-section at a fixed height \(z\). The cross-section is a right triangle with its vertices on the axes. The vertical leg on the \(y\)-axis has a length of \(1-z\), and the horizontal leg on the \(x\)-axis has a length of \(1-z\). The hypotenuse of the triangle is the line defined by the equation \(x + y = 1 - z\).
\[ A(z) = \frac{(1-z)^2}{2} = \int_0^{1-z} \int_0^{1-z-y} 1 \, dx \, dy \]