## Review for Final Exam—Part 3

1. Find the mass and center of mass of the lamina with density $\rho(x,y)=xy^2$ that occupies the region $D=\big\{ (x,y) : 0 \leq x \leq 2, -1\leq y \leq 1 \big\}.$

2. Evaluate $\displaystyle{\iiint_E xyz\, dV},$ where $E$ is the region that lies between the spheres $\rho=2, \rho=4$, and above the cone $\varphi=\pi/3.$

3. Evaluate $\displaystyle{\iint_D 4xy-y^2\, dA},$ with domain $D$ being the region bounded by $y=\sqrt{x}$ and $y=x^3.$

4. Evaluate the triple integral $\displaystyle{\iiint_E 2x\, dV},$ where $E$ is the solid bounded by the paraboloid $z=2x^2+2y^2,$ above the plane $z=8,$ and between the planes $y=0$ and $y=2.$

5. Is the vector field $\boldsymbol{F}(x,y) = \langle ye^x+\sin x\cos y, e^x + \cos x \sin y \rangle$ conservative? If it is, find a function $f(x,y)$ so that $\boldsymbol{F} =\nabla f.$

6. Evaluate the triple integral $\displaystyle{ \iiint_E \frac{1}{\sqrt{y^2+z^2+1}}\, dV},$ where $E$ is bounded by the paraboloid $y^2+z^2=1$ and the plane $x=1.$

7. Evaluate the integral $\displaystyle{\oint_C (3y^2-e^x)\, dx + (7x+\sin y)\, dy},$ where $C$ is the circle $x^2+y^2=4.$

8. Prove that the Jacobian to change from Cartesian to Spherical coordinates is $\rho^2 \sin \varphi$.

9. Find the area of the region $R$ bounded by the circles $x^2+y^2=16, x^2+y^2-4$ in the half-plane $x\geq 0.$

10. Evaluate the line integral $\displaystyle{ \int_C \big( \tfrac{x}{3} + 7y \big)\, ds},$ where $C$ is given in parametric equations by $x=3t, y=t, z=1+5t, (0 \leq t \leq 1).$
1. December 6, 2012 at 11:03 pm

Is there any way that the solutions to these practice exams can be posted after class on friday? I think it would be beneficial to know we are on the right track while working these problems out on our own.

• December 6, 2012 at 11:13 pm

Nope.

2. December 7, 2012 at 10:41 am

for 7, after applying greenes theorem to end up with double int(((7x+siny)/dx) +(3y^-e^x)/dy)dydx where do we use the circles equation in the problem?

3. December 8, 2012 at 6:19 pm

Recall that if P(x,y)=3(y^2)-(e^x) and Q(x,y)=7x+siny then we rewrite the integral and as ∫∫ (dQ/dx) – (dp/dy) dA, so you change the integral presented and change it to the integral of the derivative of 7x+siny with respect to x minus the derivative of 3(y^2)-(e^x) with respect to y on the domain D. You then use the circle, when considering the Domain. So you change to polar coordinates and integrate on the D where,θ is greater than or equal to 2π and less than or equal to 0, and r is greater than or equal than or equal to 2 and less than or equal to 0.

4. December 9, 2012 at 6:42 pm

Alright, so on number 3, I know it is a basic double integral (not cylindrical, spherical, or anything like that), and that the y limits are given, but how do we get our x limits to integrate between?

• December 9, 2012 at 11:48 pm

you need to find the intersections of y=sqrt(x) and y=x^3

• December 10, 2012 at 9:00 am

You have to graph the function and see if it’s a type 1 or type 2 domain and once you figure that out it should answer your question. Let me know if it doesn’t.

5. December 9, 2012 at 11:04 pm

For number 9 i have my two integrals from 0 to 2pi and from 2 to 4. Is that correct? Also, what are we suppose to be taking the integral of?

• December 9, 2012 at 11:55 pm

No the 0 to 2pi is incorrect. The plane x=0 is cutting the circle in half. The wording of the problem was a bit off, it should have said x>=0, so your bounds should be for only half of the circle. Remember that the area is the double integral of 1 dr (don’t forget the jacobian)

6. December 11, 2012 at 3:24 am

Do we use one of the curl equations for number 5?

• December 11, 2012 at 4:33 pm

No, you should use partial derivatives to find out whether or not it is conservative.

7. December 11, 2012 at 7:06 pm

can someone explain to me what exactly were suppose to do for number 8. Im a little confused

• December 11, 2012 at 9:20 pm

You must take the first order partial derivatives of the function and put them in a matrix to do the cross product. This formula can be found on page 1015 of the textbook. Make a three by three matrix of the variables x through z and replace them with their spherical equivalents and take the partial derivative with respect to each of the variables and find the determinant of the matrix. Let me know if this makes sense.

• December 11, 2012 at 9:27 pm

use the formula that finds the jacobian (which is located on the formula sheet as change of variables) to prove that the given one is correct.

8. December 11, 2012 at 9:34 pm

You’re supposed to do the Jacobian change of variables for spherical. In class, we were shown how the change to for polar and cylindrical change of variables works. Now we take that one set forward and do it for the spherical change. There is a bit of a difference since you will be using three variables instead two.

• December 11, 2012 at 9:35 pm

We also did the change for spherical: a whole lesson.

9. December 12, 2012 at 11:01 am

Could someone give me a hint on how to approach the triple integral in number 6?

• December 12, 2012 at 12:33 pm

change xyz into spherical coordinates then find what ro phi and theta go to and integrate the function. Dont forget the jacobian.

• December 12, 2012 at 12:34 pm

sorry that was for number 2

• December 12, 2012 at 3:18 pm

using a change of variables to cylindrical would be the easiest

10. December 12, 2012 at 2:55 pm

For number 2, is my theta between 0 and 2pi?

• December 12, 2012 at 3:19 pm

yes

11. December 12, 2012 at 6:31 pm

I’m having troubles with number 9. How would you start that off?

12. December 12, 2012 at 6:44 pm

Will curl be on the final exam at all?

13. December 12, 2012 at 6:50 pm

How do you figure out the bounds if it is above a given value/region?

14. December 12, 2012 at 8:22 pm

Can someone tell me how to start number 4?