Solution: A tank full of water has the shape of a paraboloid

Find the area of the region bounded by the given curves.

Find the area of the region bounded by the given curves.

Find the area of the region bounded by the given curves.

Find the area of the region bounded by the given curves.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volumes of the solids obtained by rotating the region bounded by the curves and about the following lines. (a) The -axis (b) The -axis (c)

Let be the region in the first quadrant bounded by the curves and . Calculate the following quantities. (a) The area of (b) The volume obtained by rotating about the -axis (c) The volume obtained by rotating about the -axis

Let be the region bounded by the curves , and . Use the Midpoint Rule with to estimate the following quantities. (a) The area of (b) The volume obtained by rotating about the -axis

Let be the region bounded by the curves and . Estimate the following quantities. (a) The -coordinates of the points of intersection of the curves (b) The area of (c) The volume generated when is rotated about the -axis (d) The volume generated when is rotated about the -axis

Each integral represents the volume of a solid. Describe the solid.

Each integral represents the volume of a solid. Describe the solid.

Each integral represents the volume of a solid. Describe the solid.

Each integral represents the volume of a solid. Describe the solid.

The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.

The base of a solid is the region bounded by the parabolas and . Find the volume of the solid if the cross-sections perpendicular to the -axis are squares with one side lying along the base.

The height of a monument is 20 m. A horizontal crosssection at a distance meters from the top is an equilateral triangle with side meters. Find the volume of the monument.

(a) The base of a solid is a square with vertices located at , and . Each cross-section perpendicular to the -axis is a semicircle. Find the volume of the solid. (b) Show that by cutting the solid of part (a), we can rearrange it to form a cone. Thus compute its volume more simply.

Find the length of the curve.

Find the length of the curve.

(a) Find the length of the curve (b) Find the area of the surface obtained by rotating the curve in part (a) about the -axis.

(a) The curve , , is rotated about the -axis. Find the area of the resulting surface. (b) Find the area of the surface obtained by rotating the curve in part (a) about the -axis.

Use Simpsons Rule with to estimate the length of the sine curve

Use Simpsons Rule with to estimate the area of the surface obtained by rotating the sine curve in Exercise 29 about the -axis.

Find the length of the curve

Find the area of the surface obtained by rotating the curve in Exercise 31 about the -axis.

A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 12 cm to 20 cm?

A 1600-pound elevator is suspended by a 200-foot cable that weighs 10 lbft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?

A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis. (a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to pump the water out of the tank. ; (b) After 4000 ft-lb of work has been done, what is the depth of the water remaining in the tank?

A trough is filled with water and its vertical ends have the shape of the parabolic region in the figure. Find the hydro - static force on one end of the trough.

A gate in an irrigation canal is constructed in the form of a trapezoid 3 ft wide at the bottom, 5 ft wide at the top, and 2 ft high. It is placed vertically in the canal so that the water just covers the gate. Find the hydrostatic force on one side of the gate.

Find the centroid of the region shown.

Find the centroid of the region bounded by the given curves.

Find the centroid of the region bounded by the given curves.

Find the volume obtained when the circle of radius 1 with center is rotated about the -axis.

Use the Theorem of Pappus and the fact that the volume of a sphere of radius is to find the centroid of the semi - circular region bounded by the curve and the -axis.

Solve the differential equation.

Solve the differential equation.

Solve the initial-value problem.

Solve the initial-value problem.

(a) Sketch a direction field for the differential equation . Then use it to sketch the four solutions that satisfy the initial conditions , , , and . (b) Check your work in part (a) by solving the differential equation explicitly. What type of curve is each solution curve?

Let be the region bounded by , , and , where . Let be the region bounded by , , and . (a) Is there a value of such that and have the same area? (b) Is there a value of such that sweeps out the same volume when rotated about the -axis and the -axis? (c) Is there a value of such that and sweep out the same volume when rotated about the -axis? (d) Is there a value of such that and sweep out the same volume when rotated about the -axis?