If the area bounded by the parabola y = x² and the line y = x is revolved about the x – axis, the volume of the solid formed may be found by using which of the following methods?

Q: 2. Find the volume generated by revolving about the y-axis, the area bounded by x² + y² = a² and x+y…

A: The bounded region: Green colored    To find: The volume of revolution around y-axis of the bounded…

Q: Suppose the following region is rotated around the y-axis to create a three dimensional solid. 4- 3-…

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Q: Show by integration that the volume V of a right circular cylinder with base radius r and height h…

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Q: If the area of the curve y = sinx from x = 0 to x = π is revolved about the y=axis i, what is the…

A: The given curve is: y=sin x      from  x=0  to  x=π

Q: If the area bounded by y = x² and y = 2 – x² is revolved about the x – axis and a vertical…

A: We have to find the volume of the solid if the area bounded by y=x2 and y=2-x2 is revolved about the…

Q: Given an area D bounded by curve y = 4x−x^2, line x+y = 6, and y-axis. Find the volume if the area D…

A: We are given an area D bounded by curve y = 4x - x2, line x + y = 6, and y-axis. We need to find the…

Q: 3. Find the volume of the solid generated when the region enclosed by y = v2x + 8, y = x, x = 0 is…

A: Given

Q: 4. Find the volume of the solid that results when the shaded region is revolved about the indicated…

A: Solution... We have volume of solid obtained by rotating the shaded region about x-axis. let…

Q: For the area bounded by y2=4x and y=x, find the volume generated if it is revolved about the x-axis

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Q: Find the volume generated by the area bounded by the parabola x=√ay , and the line y=b that is…

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Q: 2. Find the volume of the solid obtained when the region between the graph the y = f (x) = Vĩ and y…

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Q: 3. Find the volume generated if the function y =/1-x is revolved around the x-axis. (use the only…

A: If y=f(x) is a function, then the independent variable x is called the input of the function and the…

Q: 3]. , Let V = ar²h, where V the volume of a right circular cone of radiusr and height h. Show that…

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Q: Set up the equation in solving volume of the solid generated when the area bounded by x = Vy, x =…

A: The solid is generated by x=y, x=0 and y=3 around y-axis

Q: Find the volume generated if the area between y = coshx and x from x = 0 to x = 1 is resolved about…

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Q: So, the volume of the solid of revolution is 9. Volume = V = Tt (9 - x2)2 dx. To find the volume,…

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Q: If h is the height of a pyramid and A is the area of its square base, find the volume of the pyramid…

A: The square pyramid has area A=a2, the side of base would be 'a'. Consider the center of pyramid on y…

Q: Find the volume of the solid generated when the region between the graphs of the equations f(x) + x…

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Q: If the region enclosed by the y-axis, the line y=2 and the curve y=sqrtx is revolved about the…

A: Proceed as shown below...

Q: Look for the volume of the solid produced by rotating the region enclosed by about x? - 2kx – 4ky +…

A: Given curves are:  x2-2kx-4ky+k2=0 4x-4y+k=0 The area bounded by them is revolving about the line…

Q: For the area bounded by y2=4x and y=x, find the volume generated if it is revolved about the line…

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Q: Calculate the volume calculated by rotating f(x) = x2 for x ∈ [1,3] about the axis of the x.

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Q: The region bounded by the graphs of x = 0, x = VIn y, and x = V2 - In y in the first quadrant is…

A: The region bounded by the graphs x=0, x=ln y and x=2-lny in the first quadrant.we have to find…

Q: Find the volume of the solid that results when the region enclosed by x = 0, x = 9, and y: 0 is…

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Q: Deriving the formula for volume generated by revolving the first- = quadrant area bounded by the…

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Q: The base of a solid is the region in the first quadrant bounded by the line y = -1/2x + 2 and the…

A: At any point x calculate the area of cross section and integrate it over the interval.

Q: The base of a solid is in the first quadrant bounded above by y = 5 and below by y = √5x between x =…

A: The given curves are y=5, y=5x, x=0, x=5 which forms the base of a solid. The cross section of the…

Q: Find the volume in the first octant bounded above by the paraboloid z = 9 – x² – y? and on the sides…

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Q: The area bounded by the parabolas (x − 2)^2 − 9py=0 and y^2=9p(x−2) is to be revolved around x = 2.…

A: Suppose the region R is revolved about the line L, then the volume of the solid is calculated by the…

Q: A solid is constructed between the lines r = a, x = b, the graph of y = f(x) and the x-axis so that…

A: Given figure of solid with circular cross-section is ,

Q: Q2: Find the volume of the solid whose base is the region inside the curve x - y =c and y = c such…

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Q: The region in the first quadrant that is bounded above by the curvey = 1/x1/4, on the left by the…

A: Given: The region in the first quadrant that is bounded above by the curvey=1x14 on the left by the…

Q: Given a curve having an equation of y2 = 4x. a) Compute the area bounded by the curve and the line…

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Q: What is the volume of the solid whose base is the interior of the parabola y2 = x from x=0 to x=4…

A: Volume of the solid obtained by revolving y=f(x) between x=x1 and x=x2 about x-axis  is,…

Q: Find the volume when the area in y = 2x^2 − 8x + 12 and y = −2x + 12 is 9 sq units and revolves…

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Q: Find the volume of solid generated if the area bounded by each of the following curves is revolved…

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Q: 3. Find the volume generated when the region defined by y = x, y = 2x/(1+ x³) is revolved around the…

A: NOTE : As per Bartleby rules incase of multiple questions only one is answered. Please upload others…

Q: Find the volume of the solid whose base is bounded by the given function with the indicated cross…

A: To calculate the volume you have to rotate the function along the line ,x=0 which is perpendicular…

Q: Derive the equation for the volume of a sphere of radius r using the shell method.

A: Consider a circle of radius (r), which when rotated around x-axis generates a sphere.

Q: 1. Find the area bounded by the curve y = x^2  and the x-axis and the ordinates x = 0 and x = 2 2.…

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Q: What is the exact volume of a solid that is formed by rotating the region, enclosed by the functions…

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Q: 1. Show by integration that the volume V of a right circular cylinder with base radius r and height…

A: We’ll answer the first part of this question. since due to complexity. Please submit the question…

Q: Find the volume V of the solid of revolution that results when the region under the graph of f(x) =…

A: To find the Volume V.  Given that f(x) = ln(x).

Q: 2. Find the volume for the solid generated by taking the area enclosed by f(x)=cos(x), x=0, x = and…

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Q: Only one of these calculations is correct to evaluate the volume of the solid generated by revolving…

A: Here we have to set up the required integral.

Q: Find the volume of the solid bounded above by the graph of x2 + y2 +z2 = 5 and below by the graph of…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: The integral represents the volume of a solid of revolution. 2T X° dx

A: Use the principle of integration to evaluate the given problem.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.