Math 225: Calculus III   Assignment 7

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

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Instructions

This assignment contains problems involving double 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 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.

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Initialization

<<Calculus`Math225`

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

Problem 1

Consider the double integral
Integrate[x^3 y, {y, 0, 2}, {x, 4-y^2, 4}]
(The order of integration in Mathematica is from right to left so that in this example the integral with respect to x is done first and then the integral with repspect to y.) Sketch the region of integration by plotting its boundary curves and write an equivalent integral with the order of integration reversed. Evaluate both integrals and check that they agree.

Solution

Problem 2

Find the volume of the region bounded above by the hyperbolic sheet
z == 4 - x^2 + y^2, below by the xy-plane, and on the sides by the vertical planes
y == 1 and y == -1.

Solution

Problem 3

The average value of a function f over a region R in the xy-plane is defined to be the integral of f over R divided by the area of R. For this problem let R be the semicircle
x^2 + y^2 <= 1 above the x-axis. Find the average distance of the points in R to the point {0,1}, i.e., find the average value of the function
Sqrt[x^2 + (y-1)^2] over R. (Use NIntegrate[] for the integrals).

Solution

Problem 4

Find the centroid of the part of the unit disk, x^2 + y^2 <= 1, that lies in the first quadrant.

Solution

Problem 5

Find the mass and first moments about the coordinate axes of a thin triangular plate bounded by the lines y - x == 1, y + x == 1, and y == 0 in the xy-plane if the density is delta[x_,y_] = 1 - y^2.

Solution