Find the volume of the solid S described as follows. The base of S is the region bounded by 3. the curve y = x² + 2x and by the x-axis. The cross-sections perpendicular to the x-axis are equilateral triangles.

Q: Let B be the solid whose base is the circle x^2+y^2=49 and whose vertical cross sections…

A: Given, Circle- x²+y²=49=7² --------equation 1 The equation is of the form x²+y²=r². Therefore, the…

Q: Find the volume of the region bounded above by the paraboloid z = x2 + y? and below by the square R…

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Q: The base of a solid is the region between the curve, y?(7 + x) = x²(21 – x), 0 0 and the X – axis.…

A: Consider the provided question, The base of a solid is the region between the curve, y27+x=x221-x,…

Q: Use the Shell Method to find the volume of the solid obtained by rotating region under the graph of…

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Q: For the following regions R, determine which is greater--the volume of the solid generated when R is…

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Q: Find the volume of the solid that is generated when the given region is revolved as described. The…

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Q: The region bounded by y = 0 and y = kx – x is revolved around the y-axis to form a solic whose…

A: Given,        y=0, and  y=kx-x2

Q: 1. Find the volume of the solid generated by revolving the region in the first quadrant bounded by…

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Q: When P = 3 Q4 The base of a solid is the region between the curve, y?(p + x) = x²(3p – x), 0 0 and…

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Q: The base of a solid is the region between the curve, y?(p + x) = x²(3p – x), 0 < x < 3p,y 2 0 and…

A: The solution is given below

Q: Find the volume of the region under the surface z = xy? and above the area bounded by x = y? and x +…

A: Solution: The objective is to find the volume of the region

Q: ind the volume of the solid generated by revolving the region bounded by the graphs of y = x + 5 and…

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Q: Find the volume generated by revolving about the x-axis the region bounded by the following graph…

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Q: Consider the region R bounded by the graphs of y = ƒ(x) &gt; 0, x = a &gt; 0, and x = b &gt; a. If…

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Q: What values of p have the following property? The area of the region between the curve y = x-p, 1 ≤…

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Q: If you rotate the region bound by y = fx,x = h, y = 0 about the x-axis, you obtain a right- circular…

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Q: The region bounded by y = a x², x = 1, and y = 0 in quadrant I is rotated around z - axis. Given…

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Q: Given the region bounded by the graphs below, find the volume of the solid generated by revolving…

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Q: Given the volume R of the solid bounded by the graphs of z = 9- y², x +z = 9, x = 0 and N=0. a)…

A: Volume by triple integral

Q: Find the volume generated by revolving about the x-axis the region bounded by the following graph…

A: Given:- y = 1/√x , x= 1 and x=4 To find:- volume of solid formed by given curves  rotation about x…

Q: Let the region R be the area enclosed by y = 2x – 2 and x = Vy + 1. Find the volume of the solid…

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Q: The region bounded by y = Vr, y = 2, and the y-axis is revolved about the a-axis. Find the volume of…

A: The region is bounded by the curves: y=x ,     y=0    and   y axis

Q: The base of a solid is the region between the curve, y?(p + x) = x²(3p – x),0 0 and the X – axis.…

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Q: The base of the solid S is the region enclosed by the graph of y=sqrtln(x), the line x=e and the…

A: Given The base of the solid S is the region enclosed by the graph of y=sqrtln(x), the line x=e and…

Q: Consider the following region For each problem, find the volume of the solid obtained by rotating…

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Q: he region bounded by y = e, y = 0, x = 0, and x = 1 is revolved about the y- xis. Find the volume of…

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

A: Given problem:- Find the volume of the solid generated by revolving the region in the first quadrant…

Q: Let R be the region in the rst quadrant bounded by the the graph x = y2 +2 and the line y = 2. Use…

A: We first sketch the region using a graphing calculator.  The shaded region will rotate about the…

Q: Consider the region on the 1st quadrant bounded by y = √4x², x and y- axes. If the region is…

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Q: Define R as the region bounded by the functions f(x): = 3 and g(x) = x + 1 between x = 0 and x = 1.…

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Q: The base of a solid is the region between the curve, y 2 (p + x) = x 2 (3p - x), 0 0 and the X- axis…

A: Given: The base of a solid is the region between the curve y2(p+x)=x2(3p-x), 0≤x≤3p, y≥0 and the…

Q: Find the volume V of the solid below the paraboloid z = 6-x -y and above the following region.

A: We have to calculate ∬Rz dxdy

Q: Find the volume of the solid that lies under the cone z = x² + y² and above the annulus 4 < x² + y?…

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Q: 1. Find the volume of the solid obtained by rotating the region bounded by the curve y? = x?(a – x),…

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Q: Suppose that R is the finite region bounded by y = x, y = x + 2, a = 0, and a 3. Find the exact…

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Q: d the volume of the solid whose base is the region enclosed by y = x² and y = 2, and the cross…

A: Topic:- application of integration

Q: The volume beneath the surface Z=fx, y)=x-y and above the region xsysX is given by O A 1 52 OB. 54 O…

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Q: Let M be the region in the first quadrant bounded above by the graph of y = 3x and below by the…

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Q: Compute the volume of the solid above the cone o = , and between the spheres p = 1 and p = 3. Hint:…

A: We have to use spherical co ordinate system

Q: Given the volume R of the solid bounded by the graphs of z = 9 – y?, x + z = 9, x = 0 and z = 0. a)…

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Q: Find the constant k>0 such that the volume of the region lying inside the sphere 2 x +y +z =a and…

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Q: Given the region bounded by y= x^3 ,x = 2, and the x-axis (in Quadrant I) is rotated about the line…

A: Given problem:- Given the region bounded by y=x3 ,X = 2, and the x-axis (in Quadrant I) is rotated…

Q: Let R be the region enclosed by the graphs of y =e¯*,y=e", and x = 1. What is the volume of the…

A: Given- Let R be the region enclosed by the graphs of y = e-x, y = ex, and x = 1.  To find- What is…

Q: A solid is contruc lines x = a, x = b, the graph of y = f(x) and the x-axis, and the cross-sections…

A: To find the volume of the solid.

Q: Find the volume of the solid obtained by rotating the region enclosed by the curves…

A: Given problem:- Find the volume of the solid obtained by rotating the region enclosed by the curves…

Q: (c) Find the volume of the solid formed by revolving the region bounded by 4x y=e from x =0 to x=1…

A: Find the volume of the solid revolution about x axis

Q: Breble Find the volume of the solid whose base is the region inside the unit circle x2 + y? = 1, and…

A: We need to find volume.

Q: A volume is described as follows: 1. the base is the region bounded by æ = - y? + 10y + 95 and æ =…

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Q: Find SS₁² 6x d.A over the region R = {(x, y) | 0≤x≤ 2,0 ≤ y ≤ 5} by identifying it as the volume of…

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Q: Q4 The base of a solid is the region between the curve, y?(p + x) = x²(3p – x), 0 < x < 3p, y 2 0…

A: please see the next step for solution

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.