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Revolution about other axes Find the volume of the solid generated in the following situations.
49. The region R bounded by the graphs of y = sin x and y = 1 − sin x on
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- The base of a certain solid is the region between the x-axis and the curve y = sin x, between x = 0 and x = π. Each plane section of the solid perpendicular to the x-axis is an equilateral triangle with one side in the base of the solid. Find the volume of the solidarrow_forwardSet-up the double integral to find the mass of the surface S : the part of the plane z = 3 − x − 2y in the first octant, if the mass density at any point on the surface is given by δ(x, y, z) = xz with units of mass per unit area. You do not need to evaluate the double integral.arrow_forwardWork through all integrals. Determine the volumes of the solids of revolution generated by revolving the given region about the given line. Do by the method indicated. - The region bounded by y = x^2 , y = 2x, is revolved about the y axis. Do by shell method and washers.arrow_forward
- Find the volumes of the solids The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.arrow_forward3. Gabriel’s horn is formed by taking the graph of y=1/x for x ≥ 1 and rotationgit about the x-axis. (a) Find the volume of Gabriel’s Horn. (b) Set up an integral that represents the surface area of Gabriel’s Horn. (c) Show that the surface area of Gabriel’s Horn is infinite. Hint for C is attachedarrow_forward(a) sketch the solid obtained by revolving the region under the graph of f about the x-axis over the given interval, (b) describe the cross section perpendicular to the x-axis located at x, and (c) calculate the volume of the solid: f (x) = x2, [1, 3]arrow_forward
- (Integration) 8.2.1) Use the Disk Method to find the volume of a solid of revolution by rotating the region enclosed by a curve given the following by rotating around the x-axis ? y = x^2 when y = 0 to x = 2arrow_forward7.Let R be the region bounded by y= 2-x2 on the x-axis and the y axis. The solid obtained by rotating R about the line y = -2 appears below. Find the volume of the solid. Hint: dV has the shape of a washer. a= 0 b= ? f(x)= Volume = ∫ba f(x)dx =arrow_forwardc2-volume-2 Determine the volume of the solid formed by rotation about the x-axis of the region bounded by the curves y = 4x − 1 and y = 63.75 x on the interval 0 ≤ x ≤ 4.arrow_forward
- The region R is bounded by the curves x = y2 + 2, y = x - 4, and y = 0 (as shown). a. Write a single integral that gives the area of R. b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis. c. Write a single integral that gives the volume of the solid generated when R is revolved about the y-axis. d. Suppose S is a solid whose base is R and whose cross sections perpendicular to R and parallel to the x-axis are semicircles. Write a single integral that gives the volume of S.arrow_forwardA. Find the area of region S. B. Find the volume of the solid generated when R is rostered about the horizontal line y=-1. C. The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a semi-circle whose diameter lies on the base of the solid. Find the volume of this solid.arrow_forwardWork through all integrals. Determine the volumes of the solids of revolution generated by revolving the given region about the given line. Do by the method indicated. - The region bounded by y = sin(x) , y = 0, on [0,pi], is revolved about the y = 1. Do by washers.arrow_forward
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