Let R be the region in the first quadrant bounded by thecurves y= x3 and y= 2x - x2 . Calculate the followingquantities.(a) The area of R(b) The volume obtained by rotating R about the x-axis(c) The volume obtained by rotating R about the y-axis

Q: y2- 4y + 2x = 0, x = 2 and x-axis

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Q: Find the volume of the solid S generated if the region is bounded by y = sec x, the x - axis, the…

A: Given:    Region y=secx line x=π4  Volume of solid V=π∫abfx2dx   here f(x)=secx            a=0…

Q: Find the volume OF the solid obtained by rotating about the y- axis the region bounded by Y= x2 and…

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Q: Find the volume obtained by rotating the bounded region defined by y = Vx y = 0 , and x = 4 around…

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Q: Find the volume of the solid obtained when the region bounded by the curves y = x^4 − 2x and y = 0…

A: We have to find the volume of of the solid of revolution of the region bounded by the curves y=x4-2x…

Q: Find the exact value of the volume of the solid obtained by rotating the region bounded by…

A: We have to find the volume of the solid obtained by rotating the region bounded by y=5sqrt(x), x=0,…

Q: The area bounded by y = 2x² - x³, and y=0 is revolved about x=0. Find the volume of the solid…

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Q: Find the volume of the solid formed by revolving the region bounded by f (x) = 4x – x² and the…

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Q: Find the volume generated by revolving about the x-axis, the area bounded by the line 2x + 4y = 4,…

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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: What is the volume obtained by rotating the region bounded by x = (y - 3)½ and y = 2x2 + 1 around…

A: Here we use basic definition of volume .

Q: Find the volume of the solid generated by revolving about the x-axis the area bounded by y=v(x),…

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Q: The area bounded by y = x² and y = 2 – x² is revolved about the x-axis. Find the volume of the solid…

A: We have to find volume of solid revolution.

Q: Find the volume generated by revolving the area bounded by y=ex, x=0 and y=0 about the y-axis.

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Q: Set up the integral that gives the volume of the solid formed by revolving the region bounded in the…

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Q: Consider the region enclosed by y=x2 , the y-axis, and y=1. Set up the integral of the volume of the…

A: Here is the solution of the given problem:  Washers Method:  Cylindrical Method::

Q: Find the volume generated by revolving about the x-axis, the area bounded by the line 4x + 8y = 8,…

A: We need to find volume.

Q: Find the volume of the solid generated when the region bounded by y = 9x and y = 36/x is revolved…

A: The given curves fx=36xgx=9x Solving the above equation 36x=9x362x-81x2=0x362-81x=0x=0, 16

Q: Find the volume of the solid obtained by rotating the region bounded by x = -3 + y? and x = -2y…

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Q: Find the exact value of the volume of the solid obtained by rotating the region bounded by…

A: We have to find the volume of the solid obtained by rotating the region bounded by y=2sqrt(x) ,x=0…

Q: Find the volume generated by revolving the area bounded by xy=4, x=1, x=4 and the x-axis about the…

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Q: Find the volume (in cubic units) of the region bounded by the curves (y – 1)° = x – 1 ,and the lines…

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Q: Find the volume of the solid generated when the region bounded by y= 9x and y 27x is revolved about…

A: Given curves are y=9x   ....1 and y=27x  ...2

Q: Find the volume formed by rotating about the y-axis the region enclosed by: x=3y, and y3=x with y≥0.

A: Given: The curve  x=y and y3=x with y≥0. To find: The volume formed by rotating curves about the…

Q: Find the volume formed by rotating about the y-axis the region enclosed by: 3y and y3 = x with y > 0

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Q: Find the volume of the solid obtained by rotating the region under the graph of f (x) = 9 − x2 for 0…

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Q: Find the volume of the solid generated when the region bounded by y = 5x and y= 15√√x is revolved…

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Q: Find the volume of the region enclosed by z = 1 – y and z = y² – 1 for 0 < x < 42. - V =

A: To find the volume of the region enclosed by z=1-y2 and z=y2-1 for 0≤x≤42

Q: Determine the volume obtained by rotating the area enclosed by y = x(4 – x) and y 0 about the line x…

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Q: Find the volume generated when the area bounded by the two curves of x^2 = 4y and y^2 = 4x is…

A: We will find out volume of solid of revolution.

Q: Find the volume of the solid obtained if the region bounded by the graphs of y = \x , y =0, and x=9…

A: Hello. Since your question has multiple parts, we will solve the first question for you. If you want…

Q: Find the volume formed by rotating about the y-axis the region enclosed by: x = 7y and y^3= x with y…

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Q: Find the volume generated by revolving the area bounded by xy=4, x=1, x=4 and the x-axis about the…

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Q: Find the volume of the solid whose base is the region bounded between the curves y = x and y = x²,…

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Q: Find the volume of the region bounded by y = Vx and y = x rotated about the x-axis.

A: The graph is where red curve represents y=x and blue line represents y=x.

Q: find the volume of the solid obtained by rotating the region enclosed by the graphs aboutthe y-axis…

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Q: y = Vx + 1 + x and the x-axis from x = 0 to x = 3 about the y-axis.

A: When rotating about y axis, we should use shell method to find the volume. The formula for volume is…

Q: The region R in the first quadrant is bounded by y=(5x-4)^(1/2), y=0, y=4, and x=0 is revolved about…

A: Given,

Q: find the volume “V” obtained by revolutionizing the region between y=x2  and y=12-x around the axis…

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Q: Determine the volume generated by revolving the region in the first quadrant bounded by the curves…

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Q: Find the volume swept out by revolving the region bounded by the x-axis and the graph of x = 2t, y =…

A: Given that x = 2t and y = t(2–t). Obtain the interval of integration as follows.

Q: Find the volume formed by rotating the region enclosed by:   x=8y  and  y^3=x  with  y≥0   about…

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Q: Find the volume generated by revolving the area bounded by   xy=4, x=1, x=4 and the x-axis about the…

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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: find the volume of the solid obtained by rotating the region enclosed by the graphs aboutthe y-axis…

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Q: 2. Find the volume of the solid generated by revolving the region bounded by the y = ³, y= 0 and x =…

A: Find the volume of the solid generated by revolving the region bounded by the y=x3, y=0 and x=1…

Q: Find the exact volume generated by revolving the region bounded by f(x) = 2x^2 + 10 and g(x) = 4x +…

A: The volume generated by revolving the region by line y= 10 The given functions are fx=2x2+10 and…

Q: Find the volume obtained by rotating the region bounded by x = y^2 and x = y around the x-axis.

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Q: Find the volume of the solid that results when the region bounded by y = x and y = x², from x = 0 to…

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Q: Find the volume of the region bounded by y= √x and y=4x rotated about the y-axis.

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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.