6. Convert the following double integral from Cartesian to polar coordinates. Then evaluate the integral: 9-22...

 

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6. Convert the following double integral from Cartesian to polar coordinates. Then evaluate the integral: 9-22 sin(x + y) dy dr 7. Consider the region R in the xy-plane bound by the curves y = r and y = 6x-8. Let D be the solid lying above the region R and below the plane z = x. (a) Use the appropriate double integral to compute the volume of the solid D. (b) Use the appropriate triple integral to compute the volume of the solid D. (c) Find the average value of the function f(x, y) = r over the region R. 8. Consider the solid below the paraboloid z = 16-2-y and above the xy-plane. A cylindrical hole is cut through this solid using the cylinder 2 + y = 4, resulting in a new solid D. Set up a double integral in polar coordinates for computing the volume of the solid D, then compute the volume. 9. Consider the solid D bound by the sphere r+ y+z = 20 and the paraboloid z = x+ y in the first octant. Set up a triple integral in cylindrical coordinates to compute the volume of this solid. 6. Convert the following double integral from Cartesian to polar coordinates. Then evaluate the integral: 9-22 sin(x + y) dy dr 7. Consider the region R in the xy-plane bound by the curves y = r and y = 6x-8. Let D be the solid lying above the region R and below the plane z = x. (a) Use the appropriate double integral to compute the volume of the solid D. (b) Use the appropriate triple integral to compute the volume of the solid D. (c) Find the average value of the function f(x, y) = r over the region R. 8. Consider the solid below the paraboloid z = 16-2-y and above the xy-plane. A cylindrical hole is cut through this solid using the cylinder 2 + y = 4, resulting in a new solid D. Set up a double integral in polar coordinates for computing the volume of the solid D, then compute the volume. 9. Consider the solid D bound by the sphere r+ y+z = 20 and the paraboloid z = x+ y in the first octant. Set up a triple integral in cylindrical coordinates to compute the volume of this solid.