A solid lies between planes perpendicular to the y-axis at y = 0 and y = 2. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola x = √5 y^2. Find the volume of the solid. Set up the integral that gives the volume of the solid. (Type exact answers, using π as needed.) The volume of the solid is cubic units. (Type an exact answer, using π as needed.)
A solid lies between two planes perpendicular to the x-axis at x = 0 and x = 48. The cross-sections by planes perpendicular to the x-axis are circular disks whose diameters run from the line y = x/2 to the line y = x as shown in the accompanying figure. Explain why the solid has the same volume as a right circular cone with base radius 12 and height 48. The radius of a circular cross section of the solid at any value of x is. The height of the solid is Locate the right circular cone with base radius 12 and height 48 so that its vertex is at the origin and its height is along the x-axis. This cone is the surface of revolution of the function about the x-axis. (Type an equation.) The radius of the cross-section of this right circular cone at any value of x is. The height of it is. Apply Cavalieri's principle to state the conclusion. Choose the correct answer below. A. Since the radii of the cross-sections of the solids are equal, the cross-sectional areas are equal. Since the solids have equal cross-sectional areas and equal heights, the solids have the same volume by Cavalieri's principle. B. Since the base radii of the solids are equal, the areas of the bases are equal. The solids therefore have the same volume by Cavalieri's principle. C. Since the base radii of the solids are equal, the base areas are equal. Since the base areas and the heights of the solids are equal, the solids have the same volume by Cavalieri's principle. D. Since the radii of the cross-sections of the solids are equal, the cross-sectional areas are equal. Since the solids have equal cross-sectional areas, the solids have the same volume by Cavalieri's principle.
Find the volume of the solid generated by revolving the region bounded by y = 3√sin x, y = 0, and x1 = π/4 and x2 = 5π/6 about the x-axis. The volume of the solid generated by revolving the region bounded by y = 3√sin x, y = 0, and x1 = π/4 and x2 = 5π/6 about the x-axis is cubic units. (Round to the nearest hundredth.)
Find the volume of the solid generated by revolving the region about the given line. The region in the first quadrant bounded above by the line y = 2/3, below by the curve y = csc xcot x, and on the right by the line x = π/2, about the line y = 2/3. Set up the integral that gives the volume of the solid. (Type exact answers, using π as needed.) The volume of the solid generated is cubic units. (Type an exact answer, using π as needed.)
Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices (1,0), (2,1), and (1,1) about the y-axis. Use the washer method to set up the integral that gives the volume of the solid. V = dy (Type exact answers, using π as needed.) The volume of the solid generated by revolving the region enclosed by the triangle with vertices (1,0), (2,1), and (1,1) about the y-axis is cubic units. (Type an exact answer, using π as needed.)
Find the volume of the solid generated by revolving the following region about the given axis. The region in the first quadrant bounded above by the curve y = x^2, below by the x-axis, and on the right by the line x = 1, about the line x = -4. Use the washer method to set up the integral that gives the volume of the solid. Use the washer method to set up the integral that gives the volume of the solid. V = dy (Type exact answers, using π as needed.)
Use the shell method to find the volume of the solid generated by revolving the shaded region about the x-axis. Set up the integral that gives the volume of the solid. dy The volume of the solid generated by revolving the shaded region about the x-axis is cubic units. (Type an exact answer, using π as needed.)
Use the shell method to find the volume of the solid generated by revolving the region bounded by x = 5y – y^2 and x = 0 about the x-axis. The volume of the solid generated by revolving the region bounded by x = 5y – y^2 and x = 0 about the x-axis is (Type an exact answer, using π as needed)
The region in the graph shown to the right is to be revolved about the x-axis to generate a solid. There are three methods (disk, washer, and shell) to find the volume of the solid. How many integrals would be required in each method? Which of the methods is preferable for finding the volume? Explain your answer. How many integrals would be required in the disk method? How many integrals would be required in the washer method? How many integrals would be required in the shell method? Which of the following methods would be used to find the volume of the solid? A. Shell method, because the number of integrals required in this method is less than in the disk method and washer method. B. Washer method, because the number of integrals required in this method is less than in the disk method and shell method. C. Disk method, because the number of integrals required in this method is less than in the washer method and shell method.
Calculate the fluid force on one side of the plate using the coordinate system shown below. Assume the density is 62.4 lb/ft3. The fluid force on one side of the plate is Ib.
Calculate the fluid force on one side of a semicircular plate of radius 3 ft that rests vertically on its diameter at the bottom of a pool filled with water to a depth of 6ft. Assume the weight-density of water is 62.4 lb/ft3. The fluid force on one side of the plate is Ib. (Round the final answer to the nearest tenth as needed. Round all intermediate values to the nearest thousandth as needed.)
The viewing portion of the rectangular glass window in a fish tank is 54 inches wide and runs from 1.5 inches below the water's surface to 34.5 inches below the surface. Find the fluid force against this portion of the window. The weight-density of seawater is 64 lb/ft3. The fluid force against the window is lb.
A semicircle plate 10 ft in diameter sticks straight down into freshwater with the diameter along the surface. Find the force exerted by the water on one side of the plate. Assume the density is 62.4 lb/ft3. The fluid force on one side of the plate is lb. (Type an integer or a decimal rounded to the nearest tenth as needed.)
The end plates (isosceles triangles) of the trough shown to the right were designed to withstand a fluid force of 6400 lb. Assuming the density of water is 62.4 lb/ft3 , how many cubic feet can the trough hold without exceeding this limitation? What is the value of h, the depth of water that exerts a fluid force of 6400 lb? What is the integral that gives the fluid force exerted on the end of the trough when filled with water to a depth of h? The maximum volume is ft3. (Round down to the nearest cubic foot.) The value of h is ft. (Round to two decimal places as needed.)
Use the shell method to find the volume of the solid generated by revolving the region bound by y = 3x, y = 0, and x = 2 about the following lines. a. The y-axis b. The line x = 6 c. The line x = -9 d. The x-axis e. The line y = 8 f. The line y = -2 a. Set up the integral that gives the volume of the solid generated by revolving around the y-axis. The volume of the given solid is dx (Type an exact answer in terms of π.) b. Set up the integral that gives the volume of the solid generated by revolving around the line x = 6. The volume of the given solid is cubic units. (Type an exact answer in terms of π.) c. Set up the integral that gives the volume of the solid generated by revolving around the line x = -9 The volume of the given solid is cubic units. (Type an exact answer in terms of π.) d. Set up the integral that gives the volume of the solid generated by revolving around the x-axis. The volume of the given solid is cubic units. (Type an exact answer in terms of π.) e. Set up the integral that gives the volume of the solid generated by revolving around the line y = 8. The volume of the given solid is cubic units. (Type an exact answer in terms of π.) f. Set up the integral that gives the volume of the solid generated by revolving around the line y = -2. The volume of the given solid is cubic units. (Type an exact answer in terms of π.)