The volume of the solid obtained by rotating the region enclosed by y=x3, y=16x, x≥0   about the line y=0 can be computed using the method of disks or washers via an integral. The volume is V= ?  cubic units.

Q: . Find the volume of the solid obtained by rotating the region bounded by the given curves about the…

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Q: Find the volume of the solid generated by revolving the specified region about the given line. R…

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Q: Sketch the solid whose volume is given by the iterated integral and rewrite the integral using the…

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Q: Use the disc/washer or shell method to find the volume of the solid that is generated by revolving…

A: Given regions are 2x+y=2, x+2y=4, y=0

Q: Calculate the volume of the solid bounded by the cone z² = x² + y² and the cylinder x² + y? = 4 %3D

A: Solution is given below:

Q: The volume of the solid obtained by rotating the region bounded by y = x², and y = 2x about the line…

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Q: Find the volume of the region bounded above by the paraboloid z = 9 – x? - y², below by the…

A: The volume of region is calculated using the triple integral, V=∭sdV Use the cylindrical…

Q: Find the volume of the solid generated when the region enclosed by y = \2x + 8, y = x, x = 0 is…

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

Q: Express by repeated integrals, the volume of the solid bounded above by the plane 2r+ 2y- +130, on…

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Q: A metal plug has been designed using the intersection of the two cones z = √√x² + y² and √x² + y² +…

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Q: The volume of the solid obtained by rotating the region enclosed by 5x y = e° + 5, y = 0, x = 0, x =…

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Q: Find the volume of the solid generated when the region enclosed by y = Vx + 1, y = V5x, and y 0 is…

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Q: Use a triple integral to find the volume of the solid enclosed by the paraboloids y=x^2+z^2 and…

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Q: What is the volume of the solid R bounded by the x- axis, x = 1, and y = sqrt(x) when R is revolved…

A: We have,  Volume solid region  V = π∫ab(R2 - r2) dx Where, R = Outer radius is the distance from the…

Q: Use the triple integral to find the volume of the given solid. The solid enclosed by the paraboloid…

A: Use the triple integral to find the volume of the given solid. The solid enclosed by the paraboloidx…

Q: The volume of the solid obtained by rotating the region enclosed by 1 y = 0, x = 3, and x = 8, about…

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Q: Use the method of cylinders to determine the volume of the solid obtained by rotating the region…

A: Given that, y=2x+1, y=3, x=4 and the line y=10

Q: The volume of the solid obtained by rotating the region enclosed by y = 6x – 2x². y = 0 about the…

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Q: The volume of the solid obtained by rotating the region enclosed by y = x", y = 6x, about the line y…

A: First sketch the curves then use method of disc to get the volume of solid obtained.

Q: Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region…

A: Here, the region is bounded by y=6sin(πx), the x-axis, y-axis, and line x=1. The areas of the region…

Q: The volume of the solid obtained by rotating the region enclosed by 1 x² y = with limits of…

A: We have to find the volume of the solid.

Q: Use a triple integral to find the volume of the solid bounded by the graphs of the equations. z =…

A: Given equations, z=3x2,y=3-3x. The objective is to find the volume of the solid.

Q: 2. Give and evaluate a triple integral describing the volume in the first octant below the cone z =…

A: Given: Given the region in the first octant below the cone z=x2+y2 and above the paraboloid z=x2+y2…

Q: Find the volume of the solid bounded by the paraboloids 9 + x + y and z = 3 2x? – 2y - - Round your…

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Q: The volume of the solid obtained by rotating the region enclosed by y = x², Y = 2x, about the line y…

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Q: Find the volume of the solid obtained by rotating the regic y = e-*, y = 1- e-*, and x = 0 about the…

A: Given: To find the volume of the solid obtained by rotating the region bounded by the curves

Q: The area enclosed by y = 4x - x^2 and 2y = 2x + x^2 is revolved about the line x = -1. Using a…

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Q: Find the volume of the solid of revolution when the region bounded by y = x + 1, the x – axis, and…

A: Recall: Shell method,  V=2π∫abr(x)h(x)dx We have, a=0, b=2, r(x)=3-x, h(x)=x+1

Q: Find the volume of the solid bounded by the paraboloid z = 16 – x2 – y?, the cylinder x2 + y? = 8…

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Q: The region bounded by f(x) = - 3x? + 15x + 42, x = 0, and y O is rotated about the y-axis. Find the…

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Q: Find the volume of the solid obtained by rotating the region enclosed by y=e^4x+5, y=0, x=0, x=0.9…

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Q: t) The volume of the solid obtained by rotating the region bounded by y=x2,  y=3x about the line…

A: Given y=x2,  y=3x Axis of rotation line x=3 cylindrical shell method

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

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Q: The volume of the solid obtained by rotating the region enclosed by y = x², y = 3x, about the line y…

A: Given: y=x2 ; y=3x   Now find the point of intersection between the curve and line;…

Q: Use a triple integral to find the volume of the solid bounded by the graphs of the equations. z =…

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Q: The volume of the solid obtained by rotating the region enclosed by y = x², x = y? about the linex =…

A: we will solve the problem using washers method.

Q: The volume of the solid obtained by rotating the region bounded by y = a?, 6x, about the line x = 6…

A: We rotate the shaded region around x=6 lineExpress the given functions as function of y.

Q: Find the volume of the solid obtained by rotating the region enclosed by the graphs of y = 27 - x, y…

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Q: The volume of the solid obtained by rotating the region enclosed by 1 y = 0, x4 y = x = 3, and x =…

A: Plot the given curves y=1x4, y=0, x=3, x=6 on a co-ordinate plane and obtain the region enclosed by…

Q: Find the volume of the solid generated bounded by x = Vy-2, x = y? - 4y + 4, rotated about the x-…

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Q: Choose the integral that computes the volume of the solid obtained by rotating the regio by V = 4x…

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Q: Use a double integral to find the volume of the solid in the first octant lying below the plane 2x+…

A: Solution

Q: Find the volume of the solid whose base is the region bonded by y=x^2 and y= 1-x^2 and whose cross…

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Q: The volume of the solid obtained by rotating the region enclosed by y = x², y = 4x, about the line y…

A: We have to find the volume of the region.

Q: Find the volume of the Solid generated by revolving the regio x = (1 – y)? , x = 2/y and the line x…

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Q: Find the volume of the solid obtained by rotating the region bounded by x = 8y², y = 1, r = 0, about…

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Q: The volume of the solid obtained by rotating the region enclosed by y=x^3, y=16x, x≥0 about the line…

A: The given function of the disk is y = x3 and y = 16x.Thus, x3 = 16x .

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.