## Review for Final Exam—Part 3

- Find the mass and center of mass of the lamina with density that occupies the region
- Evaluate where is the region that lies between the spheres , and above the cone
- Evaluate with domain being the region bounded by and
- Evaluate the triple integral where is the solid bounded by the paraboloid above the plane and between the planes and
- Is the vector field conservative? If it is, find a function so that
- Evaluate the triple integral where is bounded by the paraboloid and the plane
- Evaluate the integral where is the circle
- Prove that the Jacobian to change from Cartesian to Spherical coordinates is .
- Find the area of the region bounded by the circles in the half-plane
- Evaluate the line integral where is given in parametric equations by

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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.

Nope.

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?

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.

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?

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

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.

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?

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)

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

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

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

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.

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.

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.

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

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

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

sorry that was for number 2

using a change of variables to cylindrical would be the easiest

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

yes

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

Will curl be on the final exam at all?

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

Can someone tell me how to start number 4?