Math 225: Calculus III   Assignment 8

Name:

Section:

I affirm that the solutions presented in this assignment are entirely my own work.

Signature:

Instructions

This assignment contains problems involving multiple integrals, see Chapter 14 in Finney & Thomas or Chapter 4 in the Lecture Notes. Before you begin, you may want to review the notebooks MultipleIntegrals, SphericalCoords and AdvancedCalculusDemo in the Notebook folder or the back of the Lecture Notes. There are also many examples of how to do these problems in the Quizzes folder.

Be sure to type in comments explaining what you are doing.

Remember to uncheck Show In/Out Names in the Kernel menu and close this group before printing.

Initialization

<<Calculus`Math225`

i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};

Problem 1

Evaluate the integral
Integrate[Cos[x^2 + y^2], {x,-3,3}, {y,0,Sqrt[9-x^2]}]
by changing to polar coordinates. (Recall that in Mathematica the order of integration is right to left so that in this example the integral with respect to y is done first and then the integral with respect to x.)

Solution

Problem 2

Find the centroid of the plane region that lies below the curve r == Sqrt[1+theta] and above the x-axis. (For this problem it is best to use NIntegrate[]).

Solution

Problem 3

Find the volume of the solid enclosed in the first octant by the parabolic sheets
y == 1 - x^2 and z == 1 - y^2.

Solution

Problem 4

Find the volume of the solid bounded below by the plane 2y + z == -2 and above by the paraboloid x^2 + (y+1)^2 + z == 8.

Solution

Problem 5

Find the volume of the solid enclosed by the spherical-coordinate surface
rho == 1 + Cos[phi].

Solution