1. Use the cylindrical shells to find the volume of the solids generated when the shaded area is revolved about the indicated axis. b) c) y=r y=4-x

Q: Find the volume of the following solid bounded by two paraboloids. laces

A: In this question, we find the volume bounded by two surfaces. Here first we use the integral to find…

Q: Apply the cylindrical shell method to obtain the volume of the following solids of revolution.

A: The volume for solid of revolution for the curve y2=x3 around the x-axis on interval 0, 1:The volume…

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

A: We know that if as semicircle is rotated about diameter then resulted figure will be sphere, whose…

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

A: On revolving the region R1 around the line Ab, the resulting solid obtained will be a cone.   This…

Q: A sphere of radius r is cut by a plane h units above the equator, where h < r. Find the volume of…

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Q: 3. Use cylindrical shells to find the volume of the solid eenerated when the shaded region is…

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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 generate by rotating the given region about the specified line. R2 about BC

A: Integrating a small volume element within appropriate limits will result in the complete volume of…

Q: B) Show that V=367 is the volume of a sphere of radius 3 by using spherical coordinates.

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Q: 2- Use the shell method to find the volume of the solids generated by revolving the shaded region…

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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: Prove that the volume of a segment of height h of a sphere of radius r is V = }nh°(3r – h)

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Q: Find the volume of the torus generated by revolving the circle (x - 2)^2 + y^2 = 1 about the y-axis.

A: Given: (x-2)2+y2=1

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 a triangle with vertices (0, 0), (0, 5), and (5, 0).…

A: Sketch the base of the solid as follows.

Q: Compute the volume of the solidbounded by the paraboloid z =x+y and the upper hemi sphere 6 -x² -…

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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: 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: Use cylindrical coordinates to find the volume of a sphere of radius a from which a central cylinder…

A: The equation for the sphere in cylindrical coordinate system is r2+z2=a2. where, b ≤ r ≤ a and…

Q: Determine the volume generated by revolving the plane figures about y axis. 4m 3m 1m 2m 4m 2m

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Q: In order to find the volume obtained by revolving the region enclosed by y = Vxe-* and y =…

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Q: Find the volume of the region in the first octant enclosed by the cylinder x^2 +y^2 =4 and the plane…

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Q: Find the volumes of the solids whose bases are bounded by the circle x^2+y^2=4, with the indicated…

A: Given, x2+y2=4 Part a: We have, x2+y2=4⇒y2=-x2+4⇒y=±4-x2 Now we find the area to use in finding the…

Q: Find the volume of the tetrahedron bounded by the four planes x=0,y=0,z=0, and 2x+2y+x=4?

A: The equation of the plane 2x+2y+z=4  can be rewritten in the form  z=4-2x-2y let z=0, 0=4-2x-2y…

Q: (9) Use the method of cylindrical shells to find the volume generated by rotating the region bounded…

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Q: Find the Volume r= 2cos 0 , x2+yナ ?= 4.

A: Topic:- volume

Q: 5. By using suitable coordinates, find the volume of the solid bounded above by the sphere a² + y2²…

A: Introduction: Evaluation of volume of a solid in three dimensional Euclidean space is an application…

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: Determine the volume generated by revolving the plane figures about line a-a. a y 3m 2m 2m 2m 2m 2m…

A: ----------------------------------------------------------------------------- Equation of line…

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: The base of a solid circular with a radius of 4 cm. Determine its volume if every plane…

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Q: Determine the volume of the region that lies under the sphere x² + y² + z² = 4, above the plane z…

A: Given: Now,

Q: Which is closest to the volume of the space in the package that is not occupied by the ball?

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Q: Find the volume of a sphere. 4/3piR^3 using cylindrical shells

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Q: If a plane region R located in the first quadrant is revolved about the x-axis, then the measure of…

A: False.

Q: use a theorem of Pappus to find the volume generated by revolving about the line x = 5 the…

A: “Pappus’s theorem for volumes:

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: Show that the surface area of a closed cuboid with square base and given volume is minimum , when it…

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Q: 9. Find the volume of the solid enclosed by the cone z = Vx? + y? and the plane z = 4.

A: we have to find the volume of the solid

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: Find the volume of the cap of a sphere with radius R and height h using a triple integral. R.

A: Consider the following figure By Pythagorean theorem x2+y2=R2 ⇒x2=R2−y2

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: Show that the volume of the region bonded by the cone z=/x +y and the paraboloid z x+y is a/6.

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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: A cone of height H with a base of radius r is cut by a plane parallel to and h units above the base,…

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Q: Show that the surface area of a closed cuboid with square base and given volume is minimum, when it…

A: Let the length be x , breadth be x and height be y of a closed cuboid with a square base.So Volume…

Q: Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of…

A: We know by the theorem of Pappu that the volume of a solid is obtained by area times the distance…

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…

Q: In Exercises 9–12, a region of the Cartesian plane is shaded. Use the Shell Method to find the…

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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.