Math 214 - Multivariable Calculus - Prep

Task 38.1

The shell method and disc method are two methods for computing the volume of a solid of revolution using a single integral. As a solid of revolution has volume, then a triple integral will give the volume, provided we can set up appropriate bounds. In this task, we'll see that the only difference between the shell and disc methods are the order in which a triple integral is done. If you've forgotten (which is completely normal), here's a reminder.

The shell-method computes volumes with $$V = \int dV = \int_a^b \underbrace{(2\pi r)(\text{height of shell at $r$})}_{\text{shell surface area = (circumference)(height)}} \underbrace{dr}_{\text{shell thickness}}.$$
The disc-method computes volumes with $$V = \int dV =\int_a^b \underbrace{\pi (\text{radius of disc at height $z$})^2}_{\text{area of disc at height $z$}} \underbrace{dz}_{\text{little height}}.$$

Consider the solid region in space that is bounded above by $z=9-x^2-y^2$ (so $z=9-r^2$) and below by the $xy$-plane. In Cartesian coordinates, the volume of this region is given by $$\int_{-3}^{3}\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}\int_{0}^{9-x^2-y^2}dzdydx.$$ This region is formed by taking the region under the parabola $z=9-r^2$ (above the plane $z=0$) and revolving it about the $z$-axis.
1. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta dzdr$.
2. Compute the two inside integrals and simplify to show that $V = \int_{0}^{3} 2\pi r (9-r^2) dr$.
3. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta drdz$. You will end up with $r=\sqrt{9-z}$ as one of the bounds.
4. Compute the two inside integrals and simplify to show that $V = \int_{0}^{9} \pi (\sqrt{9-z})^2 dz$.
5. Which order above uses the shell method, and which uses the disc method?
Consider the region in space that satisfies $0\leq a\leq r\leq b$ with $g(r)\leq z\leq f(r)$.
1. Construct a sketch of such a region. You get to pick and illustrate what $a$, $b$, $g(r)$, and $f(r)$ mean.
2. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta dzdr$.
3. Compute the two inside integrals to obtain a formula for the volume that involves a single integral in terms of $r$.
Consider the region in space that satisfies $c\leq z\leq d$ with $0\leq g(z)\leq r\leq f(z)$.
1. Construct a sketch of such a region. You get to pick and illustrate what $c$, $d$, $g(z)$, and $f(z)$ mean.
2. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta drdz$.
3. Compute the two inside integrals to obtain a formula for the volume that involves a single integral in terms of $z$.

Task 38.2

Consider the integral $\ds \int_{0}^{4}\int_{0}^{4-x} e^{(x+y)^2}dydx$ and the change-of-coordinates $u=x$, $v=x+y$.
1. Solve for $x$ and $y$ in terms of $u$ and $v$, and then compute $\frac{\partial(x,y)}{\partial(u,v)}$.
2. Set up the corresponding iterated integral using the order $dvdu$. Then set up the corresponding integral using the order $dudv$.
3. Compute the simpler of the two integrals you just set up.
Consider the integral $\ds \iint_R xy dA$ for the region $R$ that lies inside the triangle with vertices $(0,0)$, $(2,4)$, and $(3,-3)$. Notice that two of the edges of the triangle lie on the lines $y=2x$ and $y=-x$, which means we'll use the change-of-coordinates $u=2x-y$, $v=x+y$.
1. Sketch the region $R$ in the $xy$-plane. Then sketch the corresponding region in the $uv$-plane (you should obtain a triangle).
2. Set up an iterated integral using $uv$-coordinates to compute $\ds \iint_R xy dA$.
3. Compute the integral.

Task 38.3

This task focuses on exploring the curl and divergence of a vector field, using Mathematica, to gain some geometric intuition about what these vectors compute.

For each vector field below, compute the curl of the vector field, modify this chunk of Mathematica code to visualize the vector field and the curl, and look for relationships between $\vec F$ and $\vec\nabla \times \vec F$.
```
F[x_, y_, z_] := {-y, x, 0}
Show[VectorPlot3D[Evaluate[F[x, y, z]], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, VectorAspectRatio -> 1/8],
 VectorPlot3D[Evaluate[Curl[F[x, y, z], {x, y, z}]], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, VectorPoints -> Coarse, VectorAspectRatio -> 1/4]]
```
1. $\vec F(x,y,z) = (-y,x,0)$
2. $\vec F(x,y,z) = (-y,x,0)$
3. $\vec F(x,y,z) = (-z, 0, 2 x)$
4. $\vec F(x,y,z) = (2x, 3y, 4z)$
5. $\vec F(x,y,z) = (0, 3 z, -4 y)$
6. $\vec F(x,y,z) = (-z, z, x - y)$
7. $\vec F(x,y,z) = (y - z, -x + z, x - y)$
8. $\vec F(x,y,z) = (y^2, -x, 0)$
9. Pick your own vector field.
Summarize what relationships, if any, you saw.
For each vector field below, compute the divergence of the vector field, modify this chunk of Mathematica code to visualize the vector field, and look for relationships between $\vec F$ and $\vec\nabla \cdot \vec F$.
```
F[x_, y_, z_] := {2 x, 0, 0}
VectorPlot3D[Evaluate[F[x, y, z]], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]
```
1. $\vec F(x,y,z) = (2x,0,0)$
2. $\vec F(x,y,z) = (0,-3y,0)$
3. $\vec F(x,y,z) = (0,0,4z)$
4. $\vec F(x,y,z) = (2x,-3y,4z)$
5. $\vec F(x,y,z) = (x,y,z)$
6. $\vec F(x,y,z) = (-y,x,0)$
7. $\vec F(x,y,z) = (x^2,0,0)$
8. Pick your own vector field.
Summarize what relationships, if any, you saw.

Task 38.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.

Big View
Prep Day 38 Task 38.1 The shell method and disc method are two methods for computing the volume of a solid of revolution using a single integral. As a solid of revolution has volume, then a triple integral will give the volume, provided we can set up appropriate bounds. In this task, we'll see that the only difference between the shell and disc methods are the order in which a triple integral is done. If you've forgotten (which is completely normal), here's a reminder. The shell-method computes volumes with $$V = \int dV = \int_a^b \underbrace{(2\pi r)(\text{height of shell at $r$})}_{\text{shell surface area = (circumference)(height)}} \underbrace{dr}_{\text{shell thickness}}.$$ The disc-method computes volumes with $$V = \int dV =\int_a^b \underbrace{\pi (\text{radius of disc at height $z$})^2}_{\text{area of disc at height $z$}} \underbrace{dz}_{\text{little height}}.$$ Consider the solid region in space that is bounded above by $z=9-x^2-y^2$ (so $z=9-r^2$) and below by the $xy$-plane. In Cartesian coordinates, the volume of this region is given by $$\int_{-3}^{3}\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}\int_{0}^{9-x^2-y^2}dzdydx.$$ This region is formed by taking the region under the parabola $z=9-r^2$ (above the plane $z=0$) and revolving it about the $z$-axis. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta dzdr$. Compute the two inside integrals and simplify to show that $V = \int_{0}^{3} 2\pi r (9-r^2) dr$. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta drdz$. You will end up with $r=\sqrt{9-z}$ as one of the bounds. Compute the two inside integrals and simplify to show that $V = \int_{0}^{9} \pi (\sqrt{9-z})^2 dz$. Which order above uses the shell method, and which uses the disc method? Consider the region in space that satisfies $0\leq a\leq r\leq b$ with $g(r)\leq z\leq f(r)$. Construct a sketch of such a region. You get to pick and illustrate what $a$, $b$, $g(r)$, and $f(r)$ mean. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta dzdr$. Compute the two inside integrals to obtain a formula for the volume that involves a single integral in terms of $r$. Consider the region in space that satisfies $c\leq z\leq d$ with $0\leq g(z)\leq r\leq f(z)$. Construct a sketch of such a region. You get to pick and illustrate what $c$, $d$, $g(z)$, and $f(z)$ mean. Set up a triple integral in cylindrical coordinates to compute the volume of this solid using the order $d\theta drdz$. Compute the two inside integrals to obtain a formula for the volume that involves a single integral in terms of $z$. Task 38.2 Consider the integral $\ds \int_{0}^{4}\int_{0}^{4-x} e^{(x+y)^2}dydx$ and the change-of-coordinates $u=x$, $v=x+y$. Solve for $x$ and $y$ in terms of $u$ and $v$, and then compute $\frac{\partial(x,y)}{\partial(u,v)}$. Set up the corresponding iterated integral using the order $dvdu$. Then set up the corresponding integral using the order $dudv$. Compute the simpler of the two integrals you just set up. Consider the integral $\ds \iint_R xy dA$ for the region $R$ that lies inside the triangle with vertices $(0,0)$, $(2,4)$, and $(3,-3)$. Notice that two of the edges of the triangle lie on the lines $y=2x$ and $y=-x$, which means we'll use the change-of-coordinates $u=2x-y$, $v=x+y$. Sketch the region $R$ in the $xy$-plane. Then sketch the corresponding region in the $uv$-plane (you should obtain a triangle). Set up an iterated integral using $uv$-coordinates to compute $\ds \iint_R xy dA$. Compute the integral. Task 38.3 This task focuses on exploring the curl and divergence of a vector field, using Mathematica, to gain some geometric intuition about what these vectors compute. For each vector field below, compute the curl of the vector field, modify this chunk of Mathematica code to visualize the vector field and the curl, and look for relationships between $\vec F$ and $\vec\nabla \times \vec F$. F[x_, y_, z_] := {-y, x, 0} Show[VectorPlot3D[Evaluate[F[x, y, z]], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, VectorAspectRatio -> 1/8], VectorPlot3D[Evaluate[Curl[F[x, y, z], {x, y, z}]], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, VectorPoints -> Coarse, VectorAspectRatio -> 1/4]] $\vec F(x,y,z) = (-y,x,0)$ $\vec F(x,y,z) = (-y,x,0)$ $\vec F(x,y,z) = (-z, 0, 2 x)$ $\vec F(x,y,z) = (2x, 3y, 4z)$ $\vec F(x,y,z) = (0, 3 z, -4 y)$ $\vec F(x,y,z) = (-z, z, x - y)$ $\vec F(x,y,z) = (y - z, -x + z, x - y)$ $\vec F(x,y,z) = (y^2, -x, 0)$ Pick your own vector field. Summarize what relationships, if any, you saw. For each vector field below, compute the divergence of the vector field, modify this chunk of Mathematica code to visualize the vector field, and look for relationships between $\vec F$ and $\vec\nabla \cdot \vec F$. F[x_, y_, z_] := {2 x, 0, 0} VectorPlot3D[Evaluate[F[x, y, z]], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] $\vec F(x,y,z) = (2x,0,0)$ $\vec F(x,y,z) = (0,-3y,0)$ $\vec F(x,y,z) = (0,0,4z)$ $\vec F(x,y,z) = (2x,-3y,4z)$ $\vec F(x,y,z) = (x,y,z)$ $\vec F(x,y,z) = (-y,x,0)$ $\vec F(x,y,z) = (x^2,0,0)$ Pick your own vector field. Summarize what relationships, if any, you saw. Task 38.4 Pick some problems related to the topics we are discussing from the Text Book Practice page. Edit Page History Source Attach File Backlinks List Group Page last modified on November 15, 2022, at 12:38 PM
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Day 38

Task 38.1

Task 38.2

Task 38.3

Task 38.4