Find arc length and the surface area generated by rotating. x2 1 y +-;1< x < 2, around y – axis - 3 Draw a diagram to show the solid generated.

Q: Find the area of the surface generated by revolving the given curve about the x-axis. y = V4 – x, –1…

A: y=4-x2,  -1≤x≤1 Differentiating with respect to x, we get dydx=124-x20-2x=-x4-x2

Q: Find the area of the surface generated by revolving x= 4/9 - y,0sys about the y-axis. 16 x=49-v 12

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Q: Consider the given curves to do the following. y = 32 – a2, y = x² Use the method of cylindrical…

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Q: The region R enclosed by the curves y=x and y=x^2 is rotated about the line y=2. Find the volume of…

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Q: Find the area of the surface generated by revolving the given curve about the x-axis. y = V16 – x²,…

A: The given curve is y=16-x2,   -1≤x≤1. This implies that, dydx=ddx16-x2dydx=1216-x2…

Q: Use a computer algebra system to find the approximate surface area of the solid generated by…

A: To find the surface area using the computer algebra sysytem

Q: Use the shell method to find the volume of the solid generated by revolving the shaded region about…

A: It is given that, x=6-2y2⇒y=6-x2 and y=3   It is known that, volume of solid generated by revolving…

Q: Sketch the region enclosed by the curves and find its are y = cos 9x, y = 0, x = 18' 9. NOTE: Enter…

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Q: y = 3cos(x/2) the y axis and the x axis in the 1st quadrant revolved about the x-axis.

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Q: Find the area of the surface generated by revolving x=√(8y-y2) on the interval 2≤y≤5 about the…

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Q: The arc y=2 sqrt root(x) from x ∈ [4,16] is rotated about the x axis. Find the exact surface area…

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Q: Q/ Find the Surface area generated by revolving y = 1 – x² around the y- ? 0 <x <1 axis, where

A: This is a problem related to integration. Based on the general formula and concepts we will answer…

Q: Q3 Find the Surface area of the solid obtained by revolving the loop of the curve given by, x = t,…

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Q: y = 9x² and y = 0 = about x = -4

A: First we plot the graph of the region and axis of revolution.

Q: Find the volume V of the described solid S. A frustum of a right circular cone with height h, lower…

A: The volume of the required frustum can be obtained as follows.Draw the frustum as shown below.

Q: Use the cylindrical shell method to calculate the exact volume of the solid formed by rotating the…

A: Given:      fx=7sinx2 interval 0,π4 Volume of solid using cylindrical shell method is…

Q: Find the area of the surface that is generated by revolving the portion of the curve y = v16 – x…

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Q: A solid is formed by rotating the area bounded by z+y 6 and y r about the line y=-3. Using washer…

A: A solid is formed by rotating the region bounded by x2+y=6 and the line y=x about the line y=-3. The…

Q: a.) Sketch the solid obtained by revolving the region under the graph of f = x2 about the x-axis…

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Q: Sketch the region enclosed by the curves and find its area. 1 ¤ = 0, y = 1, y= e³ x = - NOTE: Enter…

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Q: Find volumc of the solid gcncrated the by revolving about the y-axis the region in 4y-y and 4-y x =

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Q: Use the cylindrical shell method to calculate the exact volume of the solid formed by rotating the…

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Q: b) Find the volume generated by revolving the region enclosed between y = -2x, x = -3, and x – axis…

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Q: The volume of the solid generated by revolving the region enclosed by the curve y=/2x ,2 < x < 4,…

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Q: A region is enclosed by the equations below. y = 2?, a = 3, y = 0 Find the volume of the solid…

A: We can solve this as follows:

Q: 63 Find the area of the surface generated by revolving x= 4/4-y, 0sys about the y-axis. x=4/4-y

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Q: The surface formed by rotating the curve y = Ex 2 1, about the x-axis is known as Gabriel's horn.…

A: the given curve is  take derivative

Q: Find the volume of the solid above the triangle with vertices (0, 0), (-1, 0) and (-1, 2), below the…

A: Given The triangle with vertices 0,0, -1,0, -1,2,below the surface z=sinπx2

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

A: When we rotate a thin vertical strip as shown in figure, about the line y=-1. We get washer with…

Q: Find the area of the surface generated by revolving x = 6t and y = 4ť (0 < t < 1) X

A: answer is in next step

Q: Find the area of the surface generated by revolving the given curve about the x-axis. y = x, 0 < x <…

A: Given: y=x, 0≤x≤2, curve about the x-axis

Q: b) Find the volume of the solid obtained when the region under the curve y= 2-x , x=0,y%3D0 revolved…

A: We will find out the required volume of solid of revolution using cylindrical shell method.

Q: Sketch the region enclosed by the curves and find its area. 1 y = 2 VI– 9x2 NOTE: Enter the cxact…

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Q: sketch the figures and show step by step solutions. Find the volume generated by the revolving the…

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Q: 1) Volume of Solid of Revolution a) about the y-axis: y = x?, y = 4 (First Quadrant) %3D

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: b)y=x²parabola, find the volume of the object formed by rotating the tangent line drawn from point…

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Q: Sketch the region enclosed by the curves and find its area. 3 x = 0, y = 1, y = e3 - = x %3D %3D…

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Q: Q/ Find the Surface area generated by revolving y = 1– x² around the y- ?0 < x <1 axis, where

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Q: Sketch the region enclosed by the curves and find its area. 1 y = 2 V1- NOTE: Enter the exact…

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Q: The surface area of the surface generated by revolving y=x³ , 0<x<1 about the x- axis is S×3.56312…

A: Option 2 none

Q: 14. Find the volume of the solid created by revolving the shaded region given below about the x axis…

A: Here given a straight line x+2y=2 and it is revolved around x axis. We need to compute its volume.If…

Q: The arc y=2 sqrt(x) from x∈[4,16] is rotated about the x axis. Find the exact surface area…

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Q: Sketch the region enclosed by the curves and find its area. 1 y = y = 2 VI– 9x2* NOTE: Enter the…

A: Calculation of Area

Q: 4. A solid is obtained by rotating about the x-axis. The curve is given to be as y = 3x2 – a³. Find…

A: Here, the solid is obtained by rotating the curve y=3x2-x3 about the x -axis as shown.

Q: Find the volume, V, of the solid formed, when the part of the curve y = 2x, is rotated about the…

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Q: (b) The volume generated by i. rotating the area enclosed between y = 1- 2² and y = 0 by the y axis.

A: Given that the area is between curves y=1-x2 and y=0. Rotated about y-axis.

Q: on the same axes, the curves y = 6- e and y= 5e coordinates of the points of intersection. Calculate…

A: Consider the given given function. y=6-ex and y=5e-x Now, draw the both graphs.

Q: Sketch the region enclosed by the curves and find its area. 3 x = - I = 0, y = 1, y= e2 NOTE: Enter…

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Q: (b) Find the volume of the solid generated by revolving the region enclosed by the curve y = 6x – x…

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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.