1,001 Calculus Practice Problems

Part I

The Questions

 

Chapter 11

Applications of Integration

This chapter presents questions related to applications of integrals: finding the area between curves, finding the volumes of a solid, and calculating the work done by a varying force. The work problems contain a variety of questions, all of which apply to a number of real-life situations and relate to questions that you may encounter in a physics class. At the end of the chapter, you answer questions related to finding the average value of a function on an interval.

The Problems You'll Work On

In this chapter, you see a variety of applications of the definite integral:

·        Finding areas between curves

·        Using the disk/washer method to find volumes of revolution

·        Using the shell method to find volumes of revolution

·        Finding volumes of solids using cross-sectional slices

·        Finding the amount of work done when applying a force to an object

·        Finding the average value of a continuous function on an interval

What to Watch Out For

Here are a few things to consider for the problems in this chapter:

·        Make graphs for the area and volume problems to help you visualize as much as possible.

·        Don't get the formulas and procedures for the disk/washer method mixed up with the shell method; it's easy to do! For example, when rotating regions about a horizontal line using disks/washers, your curve should be of the form y = f(x), but if you're using shells, your curve should be of the form x = g(y). When rotating a region about a vertical line and using disks/washers, your curve should be of the form x = g(y), but if you're using shells, your curve should be of the form y = f(x).

·        Some of the volume of revolution problems can be solved using either the disk/washer method or the shell method; other problems can be solved easily only by using one method. Pay attention to which problems seem to be doable using either method and which ones do not.

·        The work problems often give people a bit of a challenge, so don't worry if your first attempt isn't correct. Keep trying!

Areas between Curves

636–661 Find the area of the region bounded by the given curves. (Tip: It's often useful to make a rough sketch of the region.)

636. y = x2y = x4

637. y = x, 9781118496718-eq11051.eps

638. y = cos x + 1, y = x, x = 0, x = 1

639. x = y2 – y, x = 3y – y2

640. x + 1 = y29781118496718-eq11002.epsy = 0, y = 1

641. x = 1 + y2y = x – 7

642. x = y2x = 3y – 2

643. x = 2y2x + y = 1

644. y = 2xy = 8 – x2

645. x = 2 – y2x = y2 – 2

646. y = 14 – x2y = x2 – 4

647. x = y, 4x + y2 = –3

648. 9781118496718-eq11003.eps9781118496718-eq11004.eps

649. 9781118496718-eq11005.epsy = cos x, x = 0, 9781118496718-eq11006.eps

650. y = x3 – xy = 2x

651. x + y = 0, x = y2 + 4y

652. 9781118496718-eq11007.eps9781118496718-eq11008.eps

653. y = sin x, y = cos x, 9781118496718-eq11009.eps9781118496718-eq11010.eps

654. x = y29781118496718-eq11011.epsy = 0, y = 2

655. x = y2 – y, x = 4y

656. y = x – 1, y2 = 2x + 6

657. y = x, x + 2y = 0, 2x + y = 3

658. 9781118496718-eq11012.eps9781118496718-eq11013.eps

659. 9781118496718-eq11014.epsy = x2 – 3

660. y = cos x, y = sin 2x, x = 0, 9781118496718-eq11015.eps

661. y = 2e2xy = 3 – 5ex, x = 0

Finding Volumes Using Disks and Washers

662–681 Find the volume of the solid obtained by revolving the indicated region about the given line. (Tip: Making a rough sketch of the region that's being rotated is often useful.)

662. The region is bounded by the curves y = x4x = 1, and y = 0 and is rotated about the x-axis.

663. The region is bounded by the curves 9781118496718-eq11016.epsx = 0, y = 0, and y = π and is rotated about the y-axis.

664. The region is bounded by the curves 9781118496718-eq11017.epsx = 3, x = 5, and y = 0 and is rotated about the x-axis.

665. The region is bounded by the curves 9781118496718-eq11018.epsx = 1, x = 3, and y = 0 and is rotated about the x-axis.

666. The region is bounded by the curves y = csc x, 9781118496718-eq11019.eps9781118496718-eq11020.eps, and y = 0 and is rotated about the x-axis.

667. The region is bounded by the curves x + 4y = 4, x = 0, and y = 0 and is rotated about the x-axis.

668. The region is bounded by the curves x = y2 – y3 and x = 0 and is rotated about the y-axis.

669. The region is bounded by the curves 9781118496718-eq11021.epsy = 0, and x = 5 and is rotated about the x-axis.

670. The region is bounded by the curves 9781118496718-eq11022.eps and y = 2 and is rotated about the x-axis.

671. The region is bounded by the curves x = y2/3x = 0, and y = 8 and is rotated about the y-axis.

672. The region is bounded by the curves 9781118496718-eq11023.eps and y = 0 and is rotated about the x-axis.

673. The region is bounded by the curves y = sin xy = cos x, x = 0, and 9781118496718-eq11024.eps and is rotated about the x-axis.

674. The region is bounded by the curves 9781118496718-eq11025.epsy = 0, x = 0, and x = 1 and is rotated about the x-axis.

675. The region is bounded by the curves y = 3 + 2x – x2 and x + y = 3 and is rotated about the x-axis.

676. The region is bounded by the curves y = x2 and x = y2 and is rotated about the y-axis.

677. The region is bounded by the curves y = x2/3y = 1, and x = 0 and is rotated about the line y = 2.

678. The region is bounded by the curves y = x2/3y = 1, and x = 0 and is rotated about the line x = –1.

679. The region is bounded by y = sec xy = 0, and 9781118496718-eq11026.eps and is rotated about the line y = 4.

680. The region is bounded by the curves x = y2 and x = 4 and is rotated about the line x = 5.

681. The region is bounded by the curves y = exy = 0, x = 0, and x = 1 and is rotated about the line y = –1.

Finding Volume Using Cross-Sectional Slices

682–687 Find the volume of the indicated region using the method of cross-sectional slices.

682. The base of a solid C is a circular disk that has a radius of 4 and is centered at the origin. Cross-sectional slices perpendicular to the x-axis are squares. Find the volume of the solid.

683. The base of a solid C is a circular disk that has a radius of 4 and is centered at the origin. Cross-sectional slices perpendicular to the x-axis are equilateral triangles. Find the volume of the solid.

684. The base of a solid S is an elliptical region with the boundary curve 4x2 + 9y2 = 36. Cross-sectional slices perpendicular to the y-axis are squares. Find the volume of the solid.

685. The base of a solid S is triangular with vertices at (0, 0), (2, 0), and (0, 4). Cross-sectional slices perpendicular to the y-axis are isosceles triangles with height equal to the base. Find the volume of the solid.

686. The base of a solid S is an elliptical region with the boundary curve 4x2 + 9y2 = 36. Cross-sectional slices perpendicular to the x-axis are isosceles right triangles with the hypotenuse as the base. Find the volume of the solid.

687. The base of a solid S is triangular with vertices at (0, 0), (2, 0), and (0, 4). Cross-sectional slices perpendicular to the y-axis are semicircles. Find the volume of the solid.

Finding Volumes Using Cylindrical Shells

688–711 Find the volume of the region bounded by the given functions using cylindrical shells. Give an exact answer. (Tip: Making a rough sketch of the region that's being rotated is often useful.)

688. The region is bounded by the curves 9781118496718-eq11027.epsy = 0, x = 1, and x = 3 and is rotated about the y-axis.

689. The region is bounded by the curves y = x2y = 0, and x = 2 and is rotated about the y-axis.

690. The region is bounded by the curves 9781118496718-eq11028.epsx = 0, and y = 1 and is rotated about the x-axis.

691. The region is bounded by the curves y = x2, y = 0, and x = 2 and is rotated about the line x = –1.

692. The region is bounded by the curves y = 2x and y = x2 – 4x and is rotated about the y-axis.

693. The region is bounded by the curves y = x4y = 16, and x = 0 and is rotated about the x-axis.

694. The region is bounded by the curves x = 5y2 – y3 and x = 0 and is rotated about the x-axis.

695. The region is bounded by the curves y = x2 and y = 4x – x2 and is rotated about the line x = 4.

696. The region is bounded by the curves y = 1 + x + x2x = 0, x = 1, and y = 0 and is rotated about the y-axis.

697. The region is bounded by the curves y = 4x – x2x = 0, and y = 4 and is rotated about the y-axis.

698. The region is bounded by the curves 9781118496718-eq11029.epsx = 0, and y = 0 and is rotated about the x-axis.

699. The region is bounded by the curves y = 1 – x2 and y = 0 and is rotated about the line x = 2.

700. The region is bounded by the curves y = 5 + 3x – x2 and 2x + y = 5 and is rotated about the y-axis.

701. The region is bounded by the curves x + y = 5, y = x, and y = 0 and is rotated about the line x = –1.

702. The region is bounded by the curves y = sin(x2), x = 0, 9781118496718-eq11030.eps, and y = 0 and is rotated about the y-axis.

703. The region is bounded by the curves x = e y, x = 0, y = 0, and y = 2 and is rotated about the x-axis.

704. The region is bounded by the curves 9781118496718-eq11031.epsy = 0, x = 0, and x = 3 and is rotated about the y-axis.

705. The region is bounded by the curves x = y3 and y = x2 and is rotated about the line x = –1.

706. The region is bounded by the curves 9781118496718-eq11032.epsy = 0, x = 1, and x = 3 and is rotated about the line x = 4.

707. The region is bounded by the curves 9781118496718-eq11033.eps and y = x3 and is rotated about the line y = 1.

708. The region is bounded by the curves 9781118496718-eq11034.epsy = x, and y = 0 and is rotated about the x-axis.

709. The region is bounded by the curves 9781118496718-eq11035.epsy = 0, x = 0, and x = 1 and is rotated about the y-axis.

710. The region is bounded by the curves 9781118496718-eq11036.epsy = ln x, x = 1, and x = 2 and is rotated about the y-axis.

711. The region is bounded by the curves x = cos y, y = 0, and 9781118496718-eq11037.eps and is rotated about the x-axis.

Work Problems

712–735 Find the work required in each situation. Note that if the force applied is constant, work equals force times displacement (W = Fd); if the force is variable, you use the integral 9781118496718-eq11038.eps, where f (x) is the force on the object at x and the object moves from x = a to x = b.

712. In joules, how much work do you need to lift a 50-kilogram weight 3 meters from the floor? (Note: The acceleration due to gravity is 9.8 meters per second squared.)

713. In joules, how much work is done pushing a wagon a distance of 12 meters while exerting a constant force of 800 newtons in the direction of motion?

714. A heavy rope that is 30 feet long and weighs 0.75 pounds per foot hangs over the edge of a cliff. In foot-pounds, how much work is required to pull all the rope to the top of the cliff?

715. A heavy rope that is 30 feet long and weighs 0.75 pounds per foot hangs over the edge of a cliff. In foot-pounds, how much work is required to pull only half of the rope to the top of the cliff?

716. A heavy industrial cable weighing 4 pounds per foot is used to lift a 1,500-pound piece of metal up to the top of a building. In foot-pounds, how much work is required if the building is 300 feet tall?

717. A 300-pound uniform cable that's 150 feet long hangs vertically from the top of a tall building. How much work is required to lift the cable to the top of the building?

718. A container measuring 4 meters long, 2 meters wide, and 1 meter deep is full of water. In joules, how much work is required to pump the water out of the container? (Note: The density of water is 1,000 kilograms per cubic meter, and the acceleration due to gravity is 9.8 meters per second squared.)

719. If the work required to stretch a spring 2 feet beyond its natural length is 14 foot-pounds, how much work is required to stretch the spring 18 inches beyond its natural length? (Note: For a spring, force equals the spring constant k multiplied by the spring's displacement from its natural length: F(x) = kx.)

720. A force of 8 newtons stretches a spring 9 centimeters beyond its natural length. In joules, how much work is required to stretch the spring from 12 centimeters beyond its natural length to 22 centimeters beyond its natural length? Round the answer to the hundredths place. (Note: For a spring, force equals the spring constant k multiplied by the spring's displacement from its natural length: F(x) = kx. Also note that 1 newton-meter = 1 joule.)

721. A particle is located at a distance x meters from the origin, and a force of 9781118496718-eq11039.eps newtons acts on it. In joules, how much work is done moving the particle from x = 1 to x = 2? The force is directed along the x-axis. Find an exact answer. (Note: 1 newton-meter = 1 joule.)

722. Five joules of work is required to stretch a spring from its natural length of 15 centimeters to a length of 25 centimeters. In joules, how much work is required to stretch the spring from a length of 30 centimeters to a length of 42 centimeters? (Note: For a spring, force equals the spring constant k multiplied by the spring's displacement from its natural length: F(x) = kx. Also note that 1 newton-meter = 1 joule.)

723. It takes a force of 15 pounds to stretch a spring 6 inches beyond its natural length. In foot-pounds, how much work is required to stretch the spring 8 inches beyond its natural length? (Note: For a spring, force equals the spring constant k multiplied by the spring's displacement from its natural length: F(x) = kx.)

724. Suppose a spring has a natural length of 10 centimeters. If a force of 30 newtons is required to stretch the spring to a length of 15 centimeters, how much work (in joules) is required to stretch the spring from 15 centimeters to 20 centimeters? (Note: For a spring, force equals the spring constant k multiplied by the spring's displacement from its natural length: F(x) = kx. Also note that 1 newton-meter = 1 joule.)

725. Suppose a 20-foot hanging chain weighs 4 pounds per foot. In foot-pounds, how much work is done in lifting the end of the chain to the top so that the chain is folded in half?

726. A 20-meter chain lying on the ground has a mass of 100 kilograms. In joules, how much work is required to raise one end of the chain to a height of 5 meters? Assume that the chain is L-shaped after being lifted with a remaining 15 meters of chain on the ground and that the chain slides without friction as its end is lifted. Also assume that the weight density of the chain is constant and is equal to 9781118496718-eq11040.eps. Round to the nearest joule. (Note: 1 newton-meter = 1 joule.)

727. A trough has a triangular face, and the width and height of the triangle each equal 4 meters. The trough is 10 meters long and has a 3-meter spout attached to the top of the tank. If the tank is full of water, how much work is required to empty it? Round to the nearest joule. (Note: The acceleration due to gravity is 9.8 meters per second squared, and the density of water is 1,000 kilograms per cubic meter.)
9781118496718-un1101.tif

728. A cylindrical storage container has a diameter of 12 feet and a height of 8 feet. The container is filled with water to a height of 4 feet. How much work is required to pump all the water out over the side of the tank? Round to the nearest foot-pound. (Note: Water weighs 62.5 pounds per cubic foot.)
9781118496718-un1102.tif

729. A thirsty farmer is using a rope of negligible weight to pull up a bucket that weighs 5 pounds from a well that is 100 feet deep. The bucket is filled with 50 pounds of water, but as the unlucky farmer pulls up the bucket at a rate of 2 feet per second, water leaks out at a constant rate and finishes draining just as the bucket reaches the top of the well. In foot-pounds, how much work has the thirsty farmer done?

730. Ten joules of work is needed to stretch a spring from 8 centimeters to 10 centimeters. If 14 joules of work is required to stretch the spring from 10 centimeters to 12 centimeters, what is the natural length of the spring in centimeters? (Note: For a spring, force equals the spring constant k multiplied by the spring's displacement from its natural length: F(x) = kx. Also note that 1 newton-meter = 1 joule.)

731. A cylindrical storage container has a diameter of 12 feet and a height of 8 feet. The container is filled with water to a distance of 4 feet from the top of the tank. Water is being pumped out, but the pump breaks after 13,500π foot-pounds of work has been completed. In feet, how far is the remaining water from the top of the tank? Round your answer to the hundredths place.

732. Twenty-five joules of work is needed to stretch a spring from 40 centimeters to 60 centimeters. If 40 joules of work is required to stretch the spring from 60 centimeters to 80 centimeters, what is the natural length of the spring in centimeters? Round the answer to two decimal places. (Note: For a spring, force equals the spring constant k multiplied by the spring's displacement from its natural length: F(x) = kx. Also note that 1 newton-meter = 1 joule.)

733. A cylindrical storage tank with a radius of 1 meter and a length of 5 meters is lying on its side and is full of water. If the top of the tank is 3 meters below ground, how much work in joules will it take to pump all the water to ground level? (Note: The acceleration due to gravity is 9.8 meters per second squared, and the density of water is 1,000 kilograms per cubic meter.)
9781118496718-un1103.tif

734. A tank that has the shape of a hemisphere with a radius of 4 feet is full of water. If the opening to the tank is 1 foot above the top of the tank, how much work in foot-pounds is required to empty the tank?
9781118496718-un1104.tif

735. An open tank full of water has the shape of a right circular cone. The tank is 10 feet across the top and 6 feet high. In foot-pounds, how much work is done in emptying the tank by pumping the water over the top edge? Round to the nearest foot-pound. (Note: Water weighs 62.5 pounds per cubic foot.)

Average Value of a Function

736–741 Find the average value of the function on the given interval by using the formula 9781118496718-eq11041.eps.

736. f (x) = x3, [–1, 2]

737. f (x) = sin x9781118496718-eq11042.eps

738. f (x) = (sinx)(cos x), 9781118496718-eq11043.eps

739. 9781118496718-eq11044.eps, [0, 2]

740. y = sinh x cosh x, [0, ln 3]

741. 9781118496718-eq11045.eps, [1, 4]

742–747 Solve the problem using the average value formula.

742. The linear density of a metal rod measuring 8 meters in length is 9781118496718-eq11046.eps kilograms per meter, where x is measured in meters from one end of the rod. Find the average density of the rod.

743. Find all numbers d such that the average value of f (x) = 2 + 4x – 3x2 on [0, d] is equal to 3.

744. Find all numbers d such that the average value of f (x) = 3 + 6x – 9x2 on [0, d] is equal to –33.

745. Find all values of c in the given interval such that favg = f (c) for the function 9781118496718-eq11047.eps on [1, 3].

746. Find all values of c in the given interval such that favg = f (c) for the function 9781118496718-eq11048.eps on [4, 9].

747. Find all values of c in the given interval such that favg = f (c) for the function f (x) = 5 – 3x2 on [–2, 2].

748–749 Use the average value formula.

748. For the function f (x) = x sin x on the interval 9781118496718-eq11049.eps, find the average value.

749. For the function 9781118496718-eq11051.eps on the interval 9781118496718-eq11052.eps, find the average value.