Find the volume of the solid generated by revolving the region bounded by the y = 4x - x2 and y = 0 about the x-axis. Plot the graph.

Q: Find the volume of the solid generated by revolving the shaded region about the given axis. About…

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Q: Find the volume of the solid that results when the region enclosed by the curves y = -x and x = 2y“…

A: The two functions are as follows:                    y1=14x2y2=x2 The points of intersection are the…

Q: Find the volume of the solid generated by revolving the region between the parabola x = y² +1 and…

A: Given Data: Curve 1: x=y2+1 ......(1) Curve 2: x=5 ......(2) The graph of the region enclosed by the…

Q: Find the volume of the solid generated by revolving the area of a circle (x − 2)² + y² = 4; about…

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Q: Compute the volume of the solid obtained by revolving the graph of y =1+x^2/2 on [0,1] about the…

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Q: Find the volume of the solid obtained by revolving the region between the graphs of y = x² and 3x…

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Q: Find the volume of the solid formed by revolving the region bounded by y = (x – 2)2 and y = x about…

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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 of the solid formed by revolving the region bounded by the graph of x = e−y2 and the…

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Q: Find the volume of the solid generated by revolving the area bounded by the given curves about the…

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Q: Find the volume of the solid formed by rotating the region enclosed by y=x3and y=4x in the first…

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Q: Find the volume of the solid generated when the area bounded by the curve y = Vx, the y-axis, and…

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Q: Find the volume of the solid obtained by rotating the region bounded by x = -3 + y² and x = -2y…

A: Given the curve is x=-3+y2, x=-2y Outer radius=-3+y2+9=y2+6 inner radius=-2y+9 Intersection point is…

Q: Find the volume of the solid formed by revolving the region bounded by the graphs y = √x and y = x^2…

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Q: Find the volume of the solid of revolution generated by revolving the region in the first quadrant…

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Q: Find the volume of the solid that results when the region enclosed by x = y² and x = y is revolved…

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Q: Find the volume of the solid generated when the region between y = sinx and y = 0 for 0 ≤ x ≤ π is…

A: y=sinx  &  y=0  for  0≤x≤π

Q: Find the volume of the solid generated when the region bounded by y = 9 – x and y = 2x + 6 is…

A: To find the volume of solid generated by revolving around the x axis

Q: The area enclosed between the two curves x² = 2y and y² = 2x is rotated about the x- axis. Determine…

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Q: Find the volume of the solid formed by revolving the area bounded by the parabola y2 = 9x and the…

A: We want to find the area bounded by the parabola y2 = 9x and the line y = 3x about the y-axis.

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 volume of the solid generated by revolving about the x-axis, the area bounded by y = x3,…

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Q: Find the volume of the solid generated by revolving the area of a circle. (x-3)²+y²=9; about the…

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Q: 1- 0.5- x-2 tan 4 2.

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Q: Find the volume of the solid generated by revolving the region between the parabola x=y^2+1 and the…

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Q: Find the volume of the solid generated by revolving the shaded region about the given axis. About…

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Q: Find the volume (in units) of the solid generated by revolving the region bounded by the curve y = 4…

A: Use formula to find volume about x axis.

Q: Find the volume of the solid formed by rotating the area bounded by y = 1 the x- x+1 and y-axes, and…

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Q: Find the volume of the solid generated by revolving the region bounded by the y = 2x - x2 and y = 0…

A: Find out the volume of the solid generated by revolving the region by the y=2x-x2 and y=0 about the…

Q: . Find the volume of the solid formed by revolving the region bounded by the curves y = x² and y =…

A: This question is based on application of Integration.

Q: Find the volume of the solid that is generated by rotating, about the x-axis, the region bounded by…

A: Please refer the attached image for complete solution.

Q: Find the volume of the solid obtained by rotating the region across the x-axis.

A: We have to find the volume of the solid obtained by revolution the region bounded by : f(x)=ex3+e2x…

Q: Find the volume generated by rotating the region between the graphs of y = 2 - x, x = 0, and the…

A: Consider the provided region, y=2-x,x=0 Find the volume generated by rotating the region between the…

Q: Find the volume of the solid generated by revolving the region bounded by the curve y = In x, the…

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Q: . Find the volume of the solid generated by revolving the region bounded by the y = -a2 + 6x – 8 and…

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Q: Find the volume of the solid of revolution generated by revolving y = 9-x from x= - 3 to x= 3 about…

A: Given f(x)= 9-x2    where -3≤x≤3 To find the volume of the solid of revolution here use The Disc…

Q: Find the volume of the solid generated by revolving the region bounded by the graphs of y=2x2+1 and…

A: First draw the both curve and find the region which will be rotated about x-axis.than use disc…

Q: Find the volume of the solid generated by revolving the shaded region about the given axis. About…

A: Here, we need to find the volume of the solid generated by revolving the shaded region about the y -…

Q: Find the volume of the solid generated by revolving the region created by the curves, y* _y² y² 2…

A: The given curves can be plotted by:

Q: Find the volume of the solid generated by revolving the region bounded by the y = x?and y = 6x– 2x2…

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Q: Find the volume of the solid generated by revolving the region bounded by the y2-x2= 1, y = 2 about…

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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: Calculate the volume of the infinite solid obtained by rotating the region under y = (x2 + 1) −2 for…

A: Use shell method and find the volume.

Q: find the volume of the solid that results when the region enclosed by the curves x=y2 and x=y+2 are…

A: Given curves are x=y2 and x=y+2          (1) f(y)=y2 and g(y)=y+2 From (1)…

Q: Find the volume of the solid formed by rotating the region enclosed by y = elx + 3, y = 0, x = 0, x…

A: Plot the curves and find the enclosed region. The shaded region represents the enclosed region.

Q: Find the volume of the region that lies below the parabola y2 = x and above the x-axis the interval…

A: Given curves is y2=x

Q: Find the volume of the solid generated by revolving the shaded region about the y-axis. y 3

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Q: Find the volume of the solid generated in the first quadrant by revolving the region bounded by the…

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Q: Find the volume of the solid generated by revolving the unbounded region lying between y = −ln x and…

A: Given that: y=-lnx   and   the   y-axis To sketch the region:

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.