1. Use double integration in polar coordinates to find the volume of the solid that lies below the surface z = 5 + x – y and above the circle x² + y² = x in the xy-plane.

2. Find the area of the region that lies inside the cardioid r = 1 + cos θ and outside the circle r = 1 by double integration in polar coordinates.

3. Set up six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane 2x + 3y + z = 6. Evaluate one of the integrals.

4. Find the volume of the solid bounded by the graphs of these equations: the wedge cut from the cylinder x 2 + y 2 = 1 by the planes z = - y and z = 0.

5. Find the volume of the solid bounded by the graphs of these equations: z = 10 - x² - y² , y = x² , x = y² , and z = 0.

6. Find the centroid of the solid in the first octant bounded by the planes y = 0 and z = 0 and by the surfaces z = 4 – x² and x = y² .

7. Find the volume of the solid right cylinder whose base is the region in the xy – plane that lies inside the cardioid r = 1 + cos ? and outside the circle r = 1 and whose top lies in the plane z = 4.

8. Find the volume of the smaller region cut from the solid sphere p>=2 by the plane z = 1.

9. Find the volume of the region between the spheres x² + y² + z² = 1 and x² + y² + z² = 4.

10. Find the volume of the region that lies inside both the sphere x² + y² + z² = 4 and the cylinder x²+ y² = 1.

11. Find the volume of the solid within the sphere x² + y² + z² = 9 and outside the cone z=sqrt(x² + y²⁾ , and above the xy-plane.

12. Use the transformation u = 3x + 2y, v = x + 4y to evaluate the double integral of (3x²+14xy+8y²) dx dy for the region R in the first quadrant bounded by the lines y = (-3/2)x + 1, y = (-3/2) x + 3, y = (-1/4)x, and y = (-1/4)x + 1.

13. Use the transformation u = x² - y² , v = 2xy to evaluate double integral S of (x²+ y²) dx dy where S is the region bounded by x² - y²= 1, x² - y² = 9, xy = 2, and xy = 4.

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The detailed answer for the above question is provided below 1 To find the volume of the solid that lies below the surface z 5 x y and above the circle x2 y2 x in the xyplane we can use a double integ... View the full answer