Question 1

Match the equation $$\frac{x^2$}{36} + \frac{y^2$}{25} - \frac{z^2$}{4} = 1$$ with the surface it defines.

Accepted Answer

Figure 2

Question 2

Find the equation for the tangent plane to the surface $$z = \ln(6x^2$ + 9y^2$ - 2) at $$the point $$(0, 0, -2).$$

Accepted Answer

$$0 = x + y$$

Question 3

Find the absolute maximum and minimum of $$f(x, y) = 6x^2$ + 5y^2$ over the closed triangular region bounded by the lines y = x$, y = 2x$, and x = 1$.

Accepted Answer

Absolute maximum: $$117 at$$ $$(1,2)$$; absolute minimum: $$0 at$$ $$(0,0)$$

Question 4

Change the Cartesian integral $$\$$int_0$$^4 \$$int_0$$^{\sqrt{16-x^2$}} (x^2$ + y^2$)\, dy\, dx to an $$equivalent polar integral, and then evaluate.

Accepted Answer

$$64\pi$$

Question 5

Reverse the order of integration and then evaluate the integral $$\$$int_0$$^1 \int_x^1$ 16e$$^{y^2}$$ dy dx.$$

Accepted Answer

$$8(e-1)$$

Question 6

Find the flux of $$\vec{F} = x\vec{i} + y\vec{j}$$ across the boundary of the region $$G = \{(x, y) | x^2$ + y^2$ \leq 1\}.$$

Accepted Answer

$$2\pi$$

Question 7

Find the gradient field $$
abla f of $$the function $$f(x, y, z) = \frac{x^2$ y z}{xyz}.$$

Accepted Answer

$$\left(\frac{2x y z}{xyz}, \frac{x^2$ z}{xyz}, \frac{x^2$ y}{xyz}\right)$$

Question 8

Find the work done by $$F$$ over the curve in the direction of increasing $$t$$, where $$F = 3x\vec{i} + 2y\vec{j} + cz\vec{k}$$, $$C: x = \cos t, y = \sin t, z = t$$, $$0 \leq t \leq \pi.$$

Accepted Answer

$$193/24$$

Question 9

Use Stokes' Theorem to calculate the circulation of the field $$\vec{F} = 3x\vec{i} + 2y\vec{j} + z\vec{k}$$ over the cap cut from the upper hemisphere $$x^2$$ + $$y^2$$ + $$z^2 = 9$, z$$ \geq 0$$, in the indicated direction.

Accepted Answer

$$9\pi$$

Question 10

Using the Divergence Theorem, find the outward flux of $$F$$ across the boundary of the region $$D$$, where $$F = 2x\vec{i} + 2y\vec{j} + 2z\vec{k}$$, $$D is $$the region from the solid cylinder $$x^2$$ + $$y^2$$ \leq 4$$, z$ from 0$ to 2$.

Accepted Answer

$$32\pi$$

Question 11

Given the vector field $$\vec{F} = (y, -x, 0)$$, compute the circulation around the unit circle in the $$xy-$$plane using Green's Theorem.

Accepted Answer

$$2\pi$$

Question 12

Evaluate the triple integral $$\iiint_E z\, dV$$ where $$E is $$the region bounded by $$z = x^2$ + y^2$ and z = 4$.

Accepted Answer

$$8\pi$$

Question 13

A tank in the shape of a hemisphere of radius $$R is $$filled with water. Find the work required to pump all the water to the top of the tank.

Accepted Answer

$$\frac{1}{2} \pi \rho g R^4$

Question 14

Find the volume of the region bounded above by $$z = 4 - x^2$ - y^2$ and below by z = 0$.

Accepted Answer

$$8\pi$$

Question 15

Evaluate $$\$$int_0$$^1 \$$int_0$$^{\sqrt{1-x^2$}} \frac{1}{1 + x^2$ + y^2$} dy dx$$ using polar coordinates.

Accepted Answer

$$\frac{\pi}{4}$$

Question 16

Let $$f(x, y) = x^2$ + y^2$. Find the average value of f$ over the disk x^2$ + y^2$ \leq 1.$$

Accepted Answer

$$\frac{1}{2}$$

Question 17

Find the mass of a plate occupying the region $$x^2$$ + $$y^2$$ \leq 4$$ with density $$\delta(x, y) = $$x^2$$ + $$y^2.$$

Accepted Answer

$$32\pi$$

Question 18

Evaluate $$\$$int_0$$^1 \$$int_0$$^1 e$$^{x^2 + y^2}$$ dx dy$$ using an appropriate substitution.

Accepted Answer

$$\frac{e^2$ - 1}{4}$$

Question 19

A region $$R in $$the plane is bounded by $$y = x^2$ and y = 2x$. Find the area of R$.

Accepted Answer

$$\frac{8}{3}$$

Calculus III: Multivariable Calculus and Vector Calculus Study Guide

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