Revolution about other axes Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line. y = 1 - Vi, y = 1, and x = 1; about y = 1 y. y 5 1 R x5 1 y 512 Vx 1 1,
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- 7.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 =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.use definite intergrals to calculate the centroid of the region described. Uses graphs to verify that your answers are reasonable. The region between f(x)=x^3 and the line y=8 on [a,b]=[0,2]
- Find the volumes of the solids Find the volume of the solid generated by revolving about the x-axis the region bounded by y = 2 tan x, y = 0, x = -π/4, and x = π/4. (The region lies in the first and third quadrants and resembles a skewed bowtie.)Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. y=e−x, y=0, x=0, and x=ln(4); about the x-axisA. 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.
- Find the volume of the solid generated in the following situation. The region R bounded by the graph of y=3sinx and the x-axis on [0, π] is revolved about the line y=−3. The volume of the solid generated when R is revolved about the line y=−3 is:(no calculator) Consider the region R, bounded by the graphs of y=x^3, y=8 and the y-axis. The region S is bounded by y=x^3, x=2, and the x-axis (a) Find the area of region R (b) Find the volume of the solid formed by rotating region R about the y-axis (c) The region S is the base of a solid. For this solid, each cross-section perpendicular to the x-axis is a semi-circle with diameters extending from y=x^3 to the x-axis. Find the volume of this solid.MultiVariable calc: Find the volume of the solid under the plane 7x + 5y − z = 0 and above the region bounded by y = x and y = x4.
- A flat circular plate has the shape of the region x2 + y2<= 1. The plate, including the boundary where x2 + y2 = 1, is heated so that the temperature at the point (x, y) is T(x, y) = x2 + 2y2 - x. Find the temperatures at the hottest and coldest points on the plate.Y=x-4 Find the volume of the solid whem the region enclosed by x=y^2/2 and x=2 is rotated around the line x=-1Solids of revolution Let R be the region bounded by y = ln x, the x-axis, and the line x = e as shown. Find the volume of the solid that is generated when the region R is revolved about the x-axis.