A region is bounded by the x-axis, the y-axis, the line y = 2 and the curve y2 = 12 - 4x. (a.) Find the area of the region. (b.) Find the volume generated if the region is revolved about the x-axis. Also sketch the resulting

Q: Find the volume obtained by rotating the region enclosed by f(x)= 5x - x2 and g(x)= 8 - 5x + x2.…

A: Given Data: Function 1: f(x)= 5x - x2 ......(1) Function 2: g(x)= 8 - 5x + x2 ......(2) The figure…

Q: 5. Find the volume of the solid obtained by rotating the region bounded by y =x² , x= 2, and y = 0…

A: In this question we have to find the volume of the region bounded by y=14x2, x=2 and y=0 about…

Q: Let R be the region in the first quadrant bounded by the x- and Y-axes, the horizontal line y = 1,…

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Q: Draw the area bounded by the function y = x^2 + 1 and the line y = 2, and then calculate the volume…

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Q: Find the volume of the solid whose base is the region in the first quadrant bounded by y=x^2, y=1,…

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Q: Draw the region bounded by the curves. Then, use the disk method to find the volume when the region…

A: We need to draw the region bounded by the curves and then by disk method to find its volume.

Q: A torus is formed by revolving the region bounded by the circle x2 + y2 = 1 about the line x = 2 .…

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Q: 30. Find the volume obtained by rotating the region bound by the graph y = e²+e-2#, x = -2, x = 3…

A: Given that the region is bounded by y=ex+e-2x,  x=-2, x=3, y=0.And the region is rotated about…

Q: sketch the region bounded by the following curves and find the volume generated by revolving the…

A: Given the functions, y = x , x + y = 6 and x = 0 Let us draw a graph of these functions. We can see…

Q: 6. Consider the infinite region in the first quadrant bounded by the y-axis, y = 1, and 1 y =…

A:

Q: If x > 0, find the volume of the solid obtained by rotating the region enclosed by the graphs about…

A: Wed use washer method to find volume about y=-8

Q: Find the volume of the torus generated by revolving the region bounded by the graph of the circle…

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Q: The region in the first quadrant that is enclosed by the x-axis, the curve y = 5/{x√(5 - x)}, and…

A: We first sketch the region using a graphing calculator

Q: 1. Sketch the graphs of v = 2x - x2 and y = - x. Shade the region bounded by the curves, Find the…

A: Given, The sketch graph of; Y = 2x - x² Y = - x The volume of the solid that is formed by revolving…

Q: The region in the first quadrant that is enclosed by the x-axis, the curve y = 5/(xsqrt(5 - x)), and…

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Q: sketch the region bounded by the following curves and find the volume generated by revolving the…

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Q: A regular hexagonal pyramid with base edges s is inscribed in a right circular cone of radius r and…

A: A regular hexagonal pyramid that has a hexagonal base, that is, base with six equal sides and 6…

Q: Use the Pappus Theorem to find the volume of the solid formed by revolving the region enclosed by…

A: To calculate the volume of the solid formed by revolving the region enclosed by the below circle…

Q: Let M be the region in the first quadrant bounded by the graph of y = 2x + 3, the graph of y = x +…

A:

Q: Find the volume of the solid formed when the region bounded by the graph of​ f, the​ x-axis, and the…

A: Given: f(x)=x0≤x≤12x-41<x≤4-2x+154<x≤5Formula: Volume =π…

Q: a. Find the area bounded by the curves y=-x^2+4, and y = 0. b. Find the volume generated by…

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Q: Let A be the area in the first quadrant bounded by the y-axis, the line y=5 x, and the line y=2…

A: Suppose a region is enclosed by two curves, y=fx and y=gx. To calculate the volume of the region…

Q: The area bounded by the parabola ay=x², the x-axis, and the line x=b is revolved about the y-axis.…

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Q: 4- Find the volume of the solid generated by revolving the region about y-axis,the region in the…

A: Draw the graphs of y=x2, x=2 and identify the region bounded by the curves. The required region is…

Q: Draw the region bounded by the curves y=1/(x+1), y=0, x=0, and x=2. Find the volume of the region…

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Q: (a) Find the area of the region enclosed by y = lnx,the line x = e, and the x-axis. (b) Find the…

A: (a) Find the area of the region enclosed by y = lnx, the line x = e, and the x-axis. (b) Find the…

Q: The area bounded by the parabola ay = x², the x-axis, and the line x = b is revolved about the…

A:

Q: The area bounded by the parabola ay = x2, the x-axis, and the line x = b is revolved about the…

A: We can use disk and washer's method to solve this question.

Q: Find the volume of the region in the first octant bounded by the coordinate planes, the plane y+z=8,…

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Q: Determine the volume of the region enclosed by the graphs of z = f(x, y) = (x − 1)2 + y 2 and z =…

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Q: 2. Find the volume generated by rotating the given region about specified line. Sketch the region,…

A: A)  Given: x=2y;  x=0, y=4;  about…

Q: b) Find the volume generated by revolving the region x = y and y=x-2 about y-axis. (Sketch the…

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Q: A frustum of a cone is the portion of the cone bounded between the circular base and a plane…

A: Given that,  A frustum of a cone is the portion of the cone bounded between the circular base and a…

Q: Using Pappus Theorem, find the volume formed by revolving about the x-axis the area bounded in the…

A: The Pappus Theorem: If a plane area is revolved about an axis lying in its plane and not…

Q: 7. Using the circular disk method, find the volume generated by revolving about the x-axis the area…

A: Explained below

Q: The area bounded by the curve y = x3,y = 0 & x = 1, is revolved about Oy. Find the volume generated.…

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Q: Consider the infinite region in the first quadrant bounded by the graphs of y = 1 /x2 , y = 0, and x…

A: Given : The infinite region in the first quadrant bounded by the graphs of y = 1/x2 , y = 0, and x =…

Q: 3. The base of a solid is a region bounded by the lines y = 2x−2 and y=−x+4 and the x- axis. If…

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Q: The area bounded by the curve 4y = x³, the x-axis, and the line x = 2 revolves about the x-axis.…

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Q: The region bounded by the curve ×=y2, the y-axis, and the line y=2 is revolved about the x-axis.…

A: We have to find volume.

Q: Use the second Pappus Guldinus theorem to determine the volume generated by revolving the curve…

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Q: 9) Find the volume of the region bounded by y = x + 1 x +1' Y = 0, x = 0, and у %3D0, х%3D x = 4…

A: The formula for the washer method around the x-axis is V=π∫abf(x)2-g(x)2dx where f(x) is the top…

Q: a*Find the volume: Region: y=x2, y=1, x32 About y= -3

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Q: Find the volume, if possible, of the region bounded by the curves Y= 1/x^2/3, y=0 and x ≥1 revolved…

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Q: An area bounded by the curve y = sin x, the x-axis and the lines x = 0 and x = π. a. What is the…

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Q: Find the volume of the three-dimensional body caused by rotating a given straight line around an…

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Q: Find the volume generated by revolving the region bounded by y = x2, y = 0, and x = 2 about the…

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Q: The base of a three-dimensional figure is bound by the line X = √y on the interval [0, 4]. Vertical…

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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.