Find the volume of the following solids. The base of a solid is the region between the curve y = 16√ sin x and the interval [0, π] on the x-axis. The cross-sections perpendicular to the x-axis are a. equilateral triangles with bases running from the x-axis to the curve as shown in the figure. b. squares with bases running from the x-axis to the curve. a. V = (Type an exact answer, using radicals as needed.) b. V = (Type an exact answer, using radicals as needed.)
Find the volume of the solid generated by revolving the shaded region about the y-axis. The volume of the solid generated by revolving the shaded region about the y-axis is (Type an exact answer, using π as needed.)
Use the shell method to find the volume of the solid generated by revolving the shaded region about the y-axis. The volume is (Type an exact answer, using π as needed.)
It takes a force of 44, 124 lb to compress a coil spring assembly from its free height of 9 inches to its fully compressed height of 3 inches. a. What is the assembly’s force constant? b. How much work does it take to compress the assembly the first half inch? the second half inch? Answer to the nearest in-lb. a. k = lb/in b. How much work does it take to compress the assembly the first half inch? in-lb (Round to the nearest whole number as needed.) How much work does it take to compress the assembly the second half inch? in-lb (Round to the nearest whole number as needed.)
An electric elevator with a motor at the top has a multistrand cable weighing 4 lb/ft. When the car is at the first floor, 120 ft of cable are paid out, and effectively 0ft are out when the car is at the top floor. How much work does the motor do just lifting the cable when it takes the car from the first floor to the top? The amount of work required is ft - lb. (Simplify your answer.)
Calculate the fluid force on one side of a semicircular plate of radius 6ft that rests vertically on its diameter at the bottom of a pool filled with water to a depth of 8 ft. Assume the weight-density of water is 62.4 lb/ft3. The fluid force on one side of the plate is (Round the final answer to the nearest tenth as needed. Round all intermediate values to the nearest thousandth as needed.)