Exam 15: Multiple Integrals

The joint density function for random variables and is for and otherwise. Find the value of the constant . Round the answer to the nearest thousandth.

Sketch the solid whose volume is given by the iterated integral

Use the Midpoint Rule for double integrals with to estimate the area of the surface. Round your answer to three decimal places.

Find the mass of the lamina that occupies the region and has the given density function. Round your answer to two decimal places.

A cylindrical drill with radius is used to bore a hole through the center of a sphere of radius . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.

Use the given transformation to evaluate the integral. , where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and

Find the volume of the solid bounded by the surface and the planes , and . Round your answer to two decimal places.

Find the Jacobian of the transformation.

Evaluate the iterated integral by reversing the order of integration.

Use spherical coordinates to find the volume of the solid that lies within the sphere above the xy-plane and below the cone . Round the answer to two decimal places.

Find the volume of the given solid. Under the paraboloid and above the rectangle .

Find the area of the surface S where S is the part of the sphere that lies to the right of the xz-plane and inside the cylinder

A lamina occupies the part of the disk in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Find the area of the part of the plane that lies inside the cylinder .

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral , where f is a continuous function. Then write an expression for the (iterated) integral.

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

Evaluate the double integral by first identifying it as the volume of a solid.

Find the moment of inertia about the y-axis for a cube of constant density 3 and side length if one vertex is located at the origin and three edges lie along the coordinate axes.

Use the transformation to evaluate the integral , where R is the region bounded by the ellipse .

Find the center of mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola and the x-axis.