9) Given the finite region f(x)=- x=1, x=2 Vx a) Find the area of the region b) Find the volume if the region was rotated about the x-axis. c) Find the volume if the region was rotated about the line x= 2.

Q: Use the Theorem of Pappus to find the volume of thegiven solid.52. A sphere of radius r

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Q: Find the volume of the solid obtained by rotating the triangle with the vertices A(0,0), B(4,4) and…

A: Solution

Q: Let B be the solid whose base is the circle (x^2) + (y^2) = (4^2)  and whose vertical cross sections…

A: This question is based on application of Integration.

Q: Find the volume of the solid generated by revolving the given plane area having the given boundaries…

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Q: (1 point) The volume of the solid obtained by rotating the region enclosed by y = a², y = 2x, about…

A: y=x2y=2x about line x=2 for point of intersection x2=2xx2-2x=0x(x-2)=0x=0,2

Q: Use the Theorem of Pappus to find the volume of thegiven solid.A cone with height h and base radius…

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Q: The volume of the solid generated by revolving the region bounded by y = 2x - x² and (5a+1)π y = 0…

A: Volume

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: ind the volume of the solid whose base is a circular disk of radius R, centered at the origin and…

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Q: A metal plug has been designed using the intersection of the two cones z = √√x² + y² and √x² + y² +…

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Q: Find the volume of the solid whose base is bounded by the circle x2 + y2 = 16 and whose cross…

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Q: Find the volume of the solid whose base is a triangle with vertices (0, 0), (0, 5), and (5, 0).…

A: Sketch the base of the solid as follows.

Q: Find the volume of the solid generated by revolving the region bounded by the graphs of the…

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Q: Use the Theorem of Pappus to find the exact volume of a general cone created by rotating a triangle…

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Q: If the region enclosed by the y-axis, the line y=2 and the curve y=sqrtx is revolved about the…

A: Proceed as shown below...

Q: Use the Theorem of Pappus to find the exact volume of a general cone created vertices (0, 0), (1,0),…

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Q: The volume of the solid obtained by rotating the region bounded by y=4-x, y=0, X=-2, x-2 about the…

A: Determine for true or false

Q: Compute the volume lying inside the sphere x2 + y? + z2 = 46 but outside of x² + y? = 21.

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Q: Consider a right-circular cylinder of diameter 1. Form a wedge by making one slice parallel to the…

A: The diameter of the right circular cylinder is d=1. The angle which is made by other slide to the…

Q: A region bounded byf (y) = Vỹ + 1, y = 4, y = 9, and x = 1 is shown. What is the volume of the solid…

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Q: Compute the volume lying inside the sphere x2 + y? + z2 = 35 but outside of x² + y? = 19.

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Q: Derive the formula for the volume of a right circular cone of height h and radius r using an…

A: Given information:

Q: Find the volume of the solid whose base is bounded by the circle x2 + y? = 36 and whose cross…

A: Given query is to find the volume of the give solid.

Q: (c) Semicircles (d) Isosceles right triangles y 2)

A: Part C) To find the volume of the solid whose bases are bounded by the circle x2+y2=4,  And the…

Q: Show by integration that the volume of a sphere of radius r is V=4/3 πr³

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Q: Find the volume of the solid whose base is a triangle with vertices (0, 0), (0, 4), and (2, 0).…

A: Given vertices of the triangular base is 0,0, 0,4 and 2,0. We know the equation of the line passing…

Q: Find the volume of the solid generated by revolving the region enclosed by the triangle whose…

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Q: B. Find the volume generated by revolving the region enclosed by x - -y and y=x+2 about У-аxis. 20%

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Q: Use the Theorem of Pappus to find the volume of the solid of revolution The torus formed by…

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Q: Find the volume generated by rotating the given region about the specified line. R, about BC

A: In this question, we find the volume by rotating the region R1 about the line BC.

Q: What is the volume of the solid generated when the region surrounded by y = vx, y = 2 and y = 0 is…

A: Given: y=x,  y=2  and  y=0Rotation around y−axis

Q: Determine the volume of a four dimensional unit sphere: x, =r sin 8, sin 8, cos p, x, =r sin 0, sin…

A: The given problem is to find the volume of four dimensional unit sphere, we have to set up an…

Q: Find the volume of the solid generated when the region enclosed by ry = 4, x + y = 5 revolved about…

A: The volume for solid of revolution between the curves xy=4 and x+y=5 around the x-axis:The volume…

Q: Show that the rectangular box (including the top and bottom) withfixed volume V = 27 m3 and smallest…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: 1. Use the cylindrical shells to find the volume of the solids generated when the shaded area is…

A: The volume of solid of revolution can be calculated using the cylindrical elements. The formula for…

Q: What is the volume of the solid R bounded by the x- axis, x = 2, and y = x2 when R is revolved…

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Q: 4. Use cylindrical shells to find the volume of the solid generated when the region enclosed between…

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Q: Find the volume of the torus obtained by revolving the circle x^2 + y^2 = 4 about the line x = 3.

A: Find the volume of the torus obtained by revolving the circle x^2 + y^2 = 4 about the line x = 3.

Q: Find the volume of the solid of revolution bounded by the circle +y = 16, and the lines 0 and

A: To find: Volume of the solid

Q: = 3, Find the volume of the solid obtained by rotating the region bounded by y = x , y = 0, and a =…

A: In this question we have to just find the volume generated by rotation.

Q: Find the volume of the solid whose base is the region enclosed by y=x^2 and y=1, and the cross…

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Q: Find the exact volume of the solid of revolution that results when a rectangular region with…

A: Given data: When the rectangle is rotated about its length then the revolution gives the right…

Q: A hole is drilled through the center of a ball of radius r, leaving a solid with a hollow…

A: Given  that the center of a ball of radius r;  and  the  height of the cylinder be…

Q: What is the volume, in cubic units, of the solid obtained by rotating the region bounded by y = 4x-…

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Q: Compute the volume lying inside the sphere x? + y? + z? = 37 but outside of x² + y? = 21.

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Q: Find the volume of the solid that results when the region enclosed by Y = Va + 1, y = 1 and a = 1 is…

A: Here we have to find the volume of the solid that results when the region enclosed by y = √x + 1,…

Q: 1. Use Pappus' Theorem to find the volume V of the torus generated by revolving a circular region of…

A: There are two Theorem by Pappus 1st for surface area of revolution and 2nd for volume of revolution…

Q: The volume of the tetrahedron bounded by the plane * ++ = :1, {a, b, c > 0} and the coordinate a…

A: Volume of the figure will be given by integral - V = ∫∫∫dzdydxLet's find limits of z,y and x - Given…

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.