Brett Moore Assignment Section 12.3 due 09/21/2015 at 10:00pm MST 9. (1 pt) (a) Graph r = 1/(7 cos ) for /2 < < /2 and r = 1. Then write an iterated integral in polar coordinates representing the area inside the curve r = 1 and to the right of r = 1/(7 cos ). (Use t for in your work.) ,b= , With a = c= , and d = , area = ab cd d d (b) Evaluate your integral to nd the area. area = 1. (1 pt) 2 sin(x + Using polar coordinates, evaluate the integral R y2 )dA where R is the region 9 x2 + y2 64. 2. (1 pt) By changing to polar coordinates, evaluate the integral D (x 2 + y2 )5/2 dxdy The value is Choi MAT 267 ONLINE A Fall 2015 where D is the disk x2 + y2 49. . Note: You must complete part (a) in order to receive any partial credit. 3. (1 pt) Use the polar coordinates to nd the volume of a sphere of radius 8. 10. (1 pt) For each of the following, set up the integral of an arbitrary function f (x, y) over the region in whichever of rectangular or polar coordinates is most appropriate. (Use t for in your expressions.) (a) The region 4. (1 pt) Find the volume of the ellipsoid x2 + y2 + 9z2 = 16. 5. (1 pt) Find the volume of the solid enclosed by the paraboloids z = 4 x2 + y2 and z = 8 4 x2 + y2 . 6. (1 pt) A cylindrical drill with radius 5 is used to bore a hole throught the center of a sphere of radius 7. Find the volume of the ring shaped solid that remains. With a = ,b= , and d = c= integral = ab cd (b) The region , , d d d d 7. (1 pt) A volcano lls the volume between the graphs z = 0 1 , and outside the cylinder x2 + y2 = 1. Find and z = 2 + y2 )23 (x the volume of this volcano. 8. (1 pt) For the region R below, write integral in polar coordinates. With a = R f dA = ,b= b d a c ,c= R f dA as an iterated , and d = f dA, where dA = With a = ,b= c= , and d = integral = ab cd 11. (1 pt) Find the volume of the region between the graph of f (x, y) = 25 x2 y2 and the xyplane. volume = 12. (1 pt) Consider the solid shaped like an ice cream cone that is bounded by the functions z = x2 + y2 and z = 32 x2 y2 . Set up an integral in polar coordinates to nd the volume of this ice cream cone. , d , , d Instructions: Please enter the integrand in the rst answer box, typing theta for . Depending on the order of integration you Note: Use t for in your expressions. 1 choose, enter dr and dtheta in either order into the second and third answer boxes with only one dr or dtheta in each box. Then, enter the limits of integration and evaluate the integral to nd the volume. B A 14. (1 pt) Convert the integral below to polar coordinates and evaluate the integral. 3/ 2 9y2 xy dx dy 0 D y Instructions: Please enter the integrand in the rst answer box, typing theta for . Depending on the order of integration you choose, enter dr and dtheta in either order into the second and third answer boxes with only one dr or dtheta in each box. Then, enter the limits of integration and evaluate the integral to nd the volume. C A= B= C= D= B Volume = 13. (1 pt) A 7/4 f (r, ) r d dr ? 1. 4 0 Volume = 15. (1 pt) Find the volume of the wedge-shaped region (Figure 1) contained in the cylinder x2 + y2 = 9 and bounded above by the plane z = x and below by the xy-plane. 3/4 4 3/4 /2 2 5 f (r, ) r d dr ? 2. f (r, ) r dr d ? 3. 0 2 C A= B= C= D= Match each double integral in polar with the graph of the region of integration. 5 D 4 4 f (r, ) r dr d ? 4. 0 0 3/2 4 f (r, ) r dr d ? 5. 3/4 2 0 5 f (r, ) r dr d ? 6. 3/2 0 4 3/4 f (r, ) r d dr ? 7. 0 0 3/4 5 f (r, ) r dr d ? 8. /4 A 4 B C V= D E G H F 2 Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 3 \f\f