Question: Find the volume of the solid obtained
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
Transcribed Image Text:
_y = 2x, y = x²; about the x-axis %3D
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> The graphs of two functions are shown with the areas of the regions between the curves indicated. (a) What is the total area between the curves for 0 < x< 5? (b) What is the value of f [f(x) – g(x)]dx? y 27 12 4 3. 2.
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> (a) Find a positive continuous function f such that the area under the graph off from 0 to t is A(t) = t3 for all t. 0. (b) A solid is generated by rotating about the x-axis the region under the curve y – f(x), where f is a positive function and x > 0.
> If the tangent at a point P on the curve y = x3 intersects the curve again at Q, let A be the area of the region bounded by the curve and the line segment PQ. Let B be the area of the region defined in the same way starting with Q instead of P. What is t
> Suppose we are planning to make a taco from a round tortilla with diameter 8 inches by bending the tortilla so that it is shaped as if it is partially wrapped around a circular cylinder. We will fill the tortilla to the edge (but no more) with meat, chee
> Sketch the region enclosed by the given curves and find its area. у — x, у— 2x, х+у—3, х>0
> Suppose the graph of a cubic polynomial intersects the parabola y = x2 when x = 0, x = a, and x = b, where 0 < a < b. If the two regions between the curves have the same area, how is b related to a?
> A cylindrical container of radius r and height L is partially filled with a liquid whose volume is V. If the container is rotated about its axis of symmetry with constant angular speed
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> Archimedes’ Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object of density 0 floating partly submerged in a flui
> (a) Show that the volume of a segment of height h of a sphere of radius r is (b) Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the volume of one segment is twice the volume of the other, then x
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> Sketch the region enclosed by the given curves and find its area. у — 1/х, у— х, у—\x, х>0
> Find the area of the region bounded by the given curves. y = x', y = 4x – x²
> Let R1 be the region bounded by y = x2 , y = 0, and (a) Is there a value of b such that /1 and /2 have the same area? (b) Is there a value of b such that /1 sweeps out the same volume when rotated about the x-axis and the y-axis? (c) Is there a value of
> (a) Find the average value of the function f(x) – 1/sx on the interval [1, 4]. (b) Find the value c guaranteed by the Mean Value Theorem for Integrals such that fave = f(c). (c) Sketch the graph of f on [1, 4] and a rectangle whose area is the same as th
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> (a) The base of a solid is a square with vertices located at (1, 0), (0, 1), (21, 0), and (0, 21). Each cross-section perpendicular to the x-axis is a semicircle. Find the volume of the solid. (b) Show that by cutting the solid of part (a), we can rearra
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> The base of a solid is the region bounded by the parabolas y = x2 and y = 2 - x2. Find the volume of the solid if the cross-sections perpendicular to the x-axis are squares with one side lying along the base.
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> Sketch the region enclosed by the given curves and find its area. у 3 sinhx, у —е , х—0, х—2
> Each integral represents the volume of a solid. Describe the solid. 2т (6 — у)(4у — у?) dy
> Each integral represents the volume of a solid. Describe the solid. S T (2 – sin x)² dx
> Let / be the region bounded by the curves y = 1 - x2 and y = x6 - x + 1. Estimate the following quantities. (a) The x-coordinates of the points of intersection of the curves (b) The area of / (c) The volume generated when/is rotated about the x-axis (d)
> Let / be the region bounded by the curves y = tan(x2), x = 1, and y = 0. Use the Midpoint Rule with n = 4 to estimate the following quantities. (a) The area of/ b) The volume obtained by rotating / about the x-axis
> Let / be the region in the first quadrant bounded by the curves y = x3 and y = 2x 2 x2. Calculate the following quantities. (a) The area of / (b) The volume obtained by rotating / about the x-axis (c) The volume obtained by rotating / about the y-axis
> Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x2 about the following lines. (a) The x-axis (b) The y-axis (c) y = 2
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. - y = Vx, y = x²; about y = 2
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. у — соs х, [|x| < п/2, у — %: about.х — п, п/2
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. у %3 tan x, у — х, х — п/3; about the y-axis TT
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x? – y? = a², x = a + h (where a>0, h > 0); about the y-axis
> Sketch the region enclosed by the given curves and find its area. y = x*, y = 2 - |x| у — х",
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x? + 1, y = 9 – x²; about y = -1
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 0, x = 9 – y²; about x = -1
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 1 + y², y - x – 3; about the y-axis
> Find the area of the region bounded by the given curves. у 3 Vx, у— х?, х—2 y
> Find the area of the region bounded by the given curves. у у — х? — 2х sin(тx/2), 2.x y
> Find the area of the region bounded by the given curves. x + y = 0, x = y² + 3y
> Find the area of the region bounded by the given curves. у —1- 2x?, у — \x| y y = |x|
> Find the area of the region bounded by the given curves. у — x, у— —Vx, у—х— 2
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> Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.2) to the function /
> Use the diagram to show that if f is concave upward on [a, b], then a + b fave >f 2 y a +b b a 2.
> Use the result of Exercise 5.5.83 to compute the average volume of inhaled air in the lungs in one respiratory cycle.
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> Sketch the region enclosed by the given curves and find its area. y = 2x, y=x², 0<x<6
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> If f is continuous and / show that f takes on the value 4 at least once on the interval [1, 3].
> (a) Find the average value of f on the given interval. (b) Find c such that fave – f(c) (c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f. f(x) = 2xe *', [0, 2]
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> Sketch the region enclosed by the given curves and find its area. .3 y = x', y=x
> (a) Find the average value of f on the given interval. (b) Find c such that fave – f(c) (c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f. S(x) f() — (х — 3)?, [2, 5]
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> When a particle is located a distance x meters from the origin, a force of / newtons acts on it. How much work is done in moving the particle from x = 1 to x = 2? Interpret your answer by considering the work done from x = 1 to x = 1.5 and from x = 1.5 t
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