91Ó°ÊÓ

Use a double integral to compute the area of the following regions. Make a sketch of the region. The region bounded by the parabola \(y=x^{2}\) and the line \(y=4\)

Use a double integral to compute the area of the following regions. Make a sketch of the region. The region bounded by the parabola \(y=x^{2}\) and the line \(y=x+2\)

Before a gasoline-powered engine is started, water must be drained from the bottom of the fuel tank. Suppose the tank is a right circular cylinder on its side with a length of \(2 \mathrm{ft}\) and a radius of 1 ft. If the water level is 6 in above the lowest part of the tank, determine how much water must be drained from the tank.

Use a double integral to compute the area of the following regions. Make a sketch of the region. The region in the first quadrant bounded by \(y=e^{x}\) and \(x=\ln 2\)

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a solid right circular cone with height \(h\) and base radius \(r\).

Use a double integral to compute the area of the following regions. Make a sketch of the region. The region bounded by \(y=1+\sin x\) and \(y=1-\sin x\) on the interval \([0, \pi]\)

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of the cap of a sphere of radius \(R\) with thickness \(h\).

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a truncated solid cone of height \(h\) whose ends have radii \(r\) and \(R\).

Use a double integral to compute the area of the following regions. Make a sketch of the region. The region in the first quadrant bounded by \(y=x^{2}, y=5 x+6\) and \(y=6-x\)

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a solid ellipsoid with axes of length \(2 a, 2 b,\) and \(2 c\).