Exam 12: Multiple Integrals

Express the integral as an iterated integral of the form where E is the solid bounded by the surfaces

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.

Evaluate the integral by making an appropriate change of variables.Round your answer to two decimal places. R is the parallelogram bounded by the lines .

Use a triple integral to find the volume of the solid bounded by and the planes and .

Calculate the double integral.Round your answer to two decimal places.

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.

Evaluate the integral ,where R is the annular region bounded by the circles and by changing to polar coordinates.

Calculate the iterated integral.

Evaluate the double integral. , is triangular region with vertices .

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.

An agricultural sprinkler distributes water in a circular pattern of radius 100 ft.It supplies water to a depth of feet per hour at a distance of feet from the sprinkler.What is the total amount of water supplied per hour to the region inside the circle of radius 15 feet centered at the sprinkler?

A cylindrical drill with radius 3 is used to bore a hole through the center of a sphere of radius 9 .Find the volume of the ring-shaped solid that remains.Round the answer to the nearest hundredth.

Use polar coordinates to find the volume of the solid under the paraboloid and above the disk .

Find the Jacobian of the transformation.