Calculus 3 (Math-UA-123) Fall 2016 Homework 7 Due: Thursday, November 2 at the start of class Please give complete, well-written solutions to the following exercises. 1. The figure below shows the distribution of temperature, in C, in a 5 meter by 5 meter heated room. Using Riemann sums, estimate the average temperature in the room. 2. Sketch (rough sketch is ok) the solid that lies between the surface z = x2xy 2 +1 and the plane z = x + 2y and is bounded by the planes x = 0, x = 2, y = 0, and y = 4. Then, find its volume. 3. Evaluate the double integral Z Z y R x2 y 2 +1 dA, over the region R: 0 x 1, 1 y 2. 4. Sketch the region of integration (on the xy-plane) and evaluate the integral: Z 4Z y x2 y 3 dx dy. 1 y 5. (a) Sketch the region in the xy-plane that is bounded by the x-axis, y = x, and x + y = 2. (b) Express the integral of f (x, y) over this region in terms of iterated integrals in two ways. (That is, formulate the integral in two ways: in one, use dx dy; in the other, use dy dx.) (c) Using one of your answers to part (b), evaluate the integral exactly for f (x, y) = x. 6. If R is the region {(x, y) | x + y a, x2 + y 2 a2 }, with a > 0, evaluate the integral Z xy dA. R Your answer will be in terms of a. 1 Calculus 3 (Math-UA-123) Fall 2016 7. (a) Sketch the level curves of the function f (x, y) = 4 x2 2y 2 , at levels k = 4, 3, 0, and 5. (b) What region R in the xy-plane maximizes the value of Z Z (4 x2 2y 2 ) dA ? R Give reasons for your answer. (c) Then, express the double integral over the region R you specified above as an iterated integral. 8. Express D, the shaded regionZbelow, as a union of three regions of \"type I\" or \"type Z II\" and evaluate the integral y dA. D (Recall that our textbook call a region that lie between the graphs of continuous functions of x as \"type I\" and a region that lie between the graphs of continuous functions of y as \"type II\".) 2