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mathematics
calculus early transcendentals

Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions

Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply.y = x2, y = 2 - x, and x =
Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply.y = x, y = x1/3 in the
Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines.x = 2
Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines.y = 6
Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines.x = -1
Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines.y = -2
Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.y = 2
Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.y = -2
Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.x = 1
Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.x = -2
Use the shell method to find the volume of the following solids.A hole of radius r ≤ R is drilled symmetrically along the axis of a bullet. The bullet is formed by revolving the parabola about the y-axis, where 0 ≤ x ≤ R. х* y = 6( 1 R2,
Use the shell method to find the volume of the following solids.The ellipsoid formed when that part of the ellipse x2 + 2y2 = 4 with x ≥ 0 is revolved about the y-axis
Use the shell method to find the volume of the following solids.The solid formed when a hole of radius 3 is drilled symmetrically through the center of a sphere of radius 6
Use the shell method to find the volume of the following solids.The solid formed when a hole of radius 3 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9
Use the shell method to find the volume of the following solids.The solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cylinder of height 6 and radius 4
Use the shell method to find the volume of the following solids.A right circular cone of radius 3 and height 8
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = √50 - 2x2, in the first quadrant
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = √cos-1 x, in the first quadrant
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = √sin-1 x, y = √π/2, and x = 0
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = 2x-3/2, y = 2, y = 16, and x = 0
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = x3, y = 1, and x = 0
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.x = y2, x = 0, and y = 3
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.x = y2, x = 4, and y = 0
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = x, y = 2 - x, and y = 0
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 4 ,3> X = and y = V3 х- y + y3' V3+ 4 y + y3 х V3
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = 4 - x, y = 2, and x = 0 УА 4 y = 4 – x y = 2 4
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.y = 8, y = 2x + 2, x = 0, and x = 2 8 y = 2x + 2 + х 2
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = √4 - 2x2, y = 0, and x = 0, in the first quadrant
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = cos x2, y = 0, for 0 ≤ x ≤ √π/2
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = √x, y = 0, and x = 1
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = x3 - x8 + 1, y = 1
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = 1 - x2, x = 0, and y = 0, in the first quadrant
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = 3x, y = 3, and x = 0 (Use integration and check your answer using the volume formula for a cone.) УА 3 у %3D Зх х 1
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = 6 - x, y = 0, x = 2, and x = 4 УА y = 6 – x R х 2
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = (1 + x2)-1, y = 0, x = 0, and x = 2 УА y = 1 + x? R 2
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = -x2 + 4x + 2, y = x2 - 6x + 10 УА y = x2 – 6x + 10 y = -x² + 4x + 2 У 3 6. 4 х 1 4 2.
Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.y = x - x2, y = 0 у 3х — х2 1 х
Are shell method integrals easier to evaluate than washer method integrals? Explain.
Fill in the blanks: A region R is revolved about the x-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to ________ or using the shell method and integrating with respect to ________.
Fill in the blanks: A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to ________ or using the shell method and integrating with respect to ________.
Assume f and g are continuous with f(x) ≥ g(x) on [a, b]. The region bounded by the graphs of f and g and the lines x = a and x = b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.
A solid wooden object turned on a lathe has a length of 50 cm and diameters (measured in cm) shown in the figure. (A lathe is a tool that spins and cuts a block of wood so that it has circular cross sections.) Use left Riemann sums with uniformly spaced grid points to estimate the volume of the
Sketch a solid of revolution whose volume by the disk method is given by the following integrals. Indicate the function that generates the solid. Solutions are not unique.a.b. TT T sin? x dx 7 (x? + 2х + 1) dx TT
Let R be the region bounded by the curve y = √x + a (with a > 0), the y-axis, and the x-axis. Let S be the solid generated by rotating R about the y-axis. Let T be the inscribed cone that has the same circular base as S and height √a. Show that volume(S)/volume(T) = 8/5.
Find the volume of the solid of revolution. Sketch the region in question.The region bounded by y = e-x, y = 0, x = 0, and x = p > 0 revolved about the x-axis (Is the volume bounded as p→∞)
Find the volume of the solid of revolution. Sketch the region in question.The region bounded by y = ln x, y = ln x2, and y = ln 8 revolved about the y-axis
Find the volume of the solid of revolution. Sketch the region in question.The region bounded by y = e-x, y = ex, x = 0, and x = ln 4 revolved about the x-axis
Find the volume of the solid of revolution. Sketch the region in question.The region bounded by y = ex, y = 0, x = 0, and x = 2 revolved about the x-axis
Find the volume of the solid of revolution. Sketch the region in question.The region bounded by revolved about the x-axis and y x² + 1 /2 Vx?
Find the volume of the solid of revolution. Sketch the region in question.The region bounded by y = 1/√x, y = 0, x = 2, and x = 6 revolved about the x-axis
Find the volume of the solid of revolution. Sketch the region in question.The region bounded by y = (ln x)/√x, y = 0, and x = 2 revolved about the x-axis
Determine whether the following statements are true and give an explanation or counterexample.a. A pyramid is a solid of revolution.b. The volume of a hemisphere can be computed using the disk method.c. Let R1 be the region bounded by y = cos x and the x-axis on [-π/2, π/2]. Let R2 be the region
Find the volume of the solid generated in the following situations.The region R is bounded by the graph of f(x) = 2x(2 - x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y = 2 or the volume of the solid generated when R is revolved about the
Find the volume of the solid generated in the following situations.The region R in the first quadrant bounded by the graphs of y = 2 - x and y = 2 - 2x is revolved about the line x = 3.
Find the volume of the solid generated in the following situations.The region R in the first quadrant bounded by the graphs of y = x and y = 1 + x/2 is revolved about the line y = 3.
Find the volume of the solid generated in the following situations.The region R bounded by the graphs of y = sin x andis revolved about the line y = -1. TT 5T y = 1 – sin x on 6’ 6
Find the volume of the solid generated in the following situations.The region R bounded by the graph of y = ln x and the y-axis on the interval 0 ≤ y ≤ 1 is revolved about the line x = -1.
Find the volume of the solid generated in the following situations.The region R bounded by the graph of y = 2 sin x and the x-axis on [0, π] is revolved about the line y = -2.
Find the volume of the solid generated in the following situations.The region R bounded by the graphs of x = 0, y = √x, and y = 2 is revolved around the line x = 4.
Find the volume of the solid generated in the following situations.The region R bounded by the graphs of x = 0, y = √x, and y = 1 is revolved around the line y = 1.
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.R is bounded by y = x2 and y = √8x.
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.R is bounded by y = 1 - x3, the x-axis, and the y-axis.
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.R is bounded by y = 4 - 2x, the x-axis, and the y-axis.
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.R is bounded by y = 2x, the x-axis, and x = 5.
Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis.y = sin-1 x, x = 0, y = π/4
Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis.x = √4 - y2, x = 0
Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis.y = √x, y = 0, x = 4
Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis.y = x3, y = 0, x = 2
Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis.y = 0, y = ln x, y = 2, x = 0 y = 2 2 - y = In x 1 х
Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis.y = x, y = 2x, y = 6 УА 2x y = 6 у 3 х х
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = |x|, y = 2 - x2
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = sin x, y = √sin x, for 0 ≤ x ≤ π/2
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = √sin x, y = 1, x = 0
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = x + 3, y = x2 + 1
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = x, y = x + 2, x = 0, x = 4 y, у %3Dх+ 2 у 3 х х 4.
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = ex/2, y = e-x/2, x = ln 2, x = ln 3 УА y = ex/2 y = e¯x/2 In 2 In 3
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = x, y = 4√x У y = у %3 х х
Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis.y = x, y = 2√x УА y = 2Vx у — х R х
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = 0, x = -1/2, and x = 1/2 y = VI – x² .2
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis. y = 0, x = -1, and x = 1 уз V1 + x2 .2
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = sec x, y = 0, x = 0, and x = π/4
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = 0, x = 0, and x = 1/2 y = 0, x = 0, and x y : VI %3D
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = √25 - x2, y = 0 (Verify that your answer agrees with the volume formula for a sphere.)
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = sin x on [0, π], y = 0 (Recall that sin2 x = 1/2 (1 - cos 2x).)
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = cos x on [0, π/2], y = 0, x = 0 (Recall that cos2 x = 1/2 (1 + cos 2x).) УА 1 y = cos x R х
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = e-x, y = 0, x = 0, x = ln 4 y = e=x R In 4 х
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = 2 - 2x, y = 0, x = 0 (Verify that your answer agrees with the volume formula for a cone.) y = 2 – 2x
Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.y = 2x, y = 0, x = 3 (Verify that your answer agrees with the volume formula for a cone.) (3, 6) у %3D 2х R х 3
Use the general slicing method to find the volume of the following solids.A circular cylinder of radius r and height h whose axis is at an angle of π/4 to the base circular base т
Use the general slicing method to find the volume of the following solids.The tetrahedron (pyramid with four triangular faces), all of whose edges have length 4
Use the general slicing method to find the volume of the following solids.The pyramid with a square base 4 m on a side and a height of 2 m (Use calculus.)
The solid whose base is the triangle with vertices (0, 0), (2, 0), and (0, 2), and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles
The solid whose base is the region bounded by y = x2 and the line y = 1, and whose cross sections perpendicular to the base and parallel to the x-axis are squares square cross section base y = x2
The solid with a semicircular base of radius 5 whose cross sections perpendicular to the base and parallel to the diameter are squares
The solid with a circular base of radius 5 whose cross sections perpendicular to the base and parallel to the x-axis are equilateral triangles equilateral triangles circular base х
The solid whose base is the region bounded by the curve y = √cos x and the x-axis on [-π/2, π/2], and whose cross sections through the solid perpendicular to the x-axis are isosceles right triangles with a horizontal leg in the xy-plane and a vertical leg above the x-axis y y = Vcos x х х
The solid whose base is the region bounded by the semicircle y = √1 - x2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares y = /1 – x² х
Use the general slicing method to find the volume of the following solids.The solid whose base is the region bounded by the curves y = x2 and y = 2 - x2, and whose cross sections through the solid perpendicular to the x-axis are squares y = x2 y = 2 – x2 х y
The region R bounded by the graph of y = f(x) ≥ 0 and the x-axis on [a, b] is revolved about the line y = -2 to form a solid of revolution whose cross sections are washers. What are the inner and outer radii of the washer at a point x in [a, b]?
Why is the disk method a special case of the general slicing method?

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