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

Q: Find the volume of the region U defined by the inequalities 0≤z≤2,0≤y+z≤5,0≤x+y+z≤10.

A: Obviously, this region is a parallelopiped. It’s convenient to change variables in such a way to…

Q: Find the area of the circular washer with outerradius 2 and inner radius 1, using Fubini’s Theorem

A: Given Circle with radius ‘2’  Circle with radius ‘1’

Q: Find the volume of tetrahedron with vertices (0,0,0),(1,0,0),(0,1,0)and (0,0,2) by triple integral?…

A: We have to find the volume of the tetrahedrane with vertices 0,0,0, 1,0,0, 0,1,0 and 0,0,2 by triple…

Q: Calculate the volume of one of the wedge-like objects formed by cutting the cylinder x²+9y²=a² with…

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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: 2. Suppose that the triangle with vertices (-1,-1), (0, 1) and (1,–1) is revolved about the y –…

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Q: Calculate the volume of the solid that is limited superiorly by the sphere x2+y2+z2=9 and inferiorly…

A: To Calculate: Volume of the solid bounded superiorly by x2+y2+z2=9 and inferiorly by x2+y2=z2.

Q: Find the radius and height of a cylindrical can of total surface area A whose volume is as large as…

A: Find the radius and height of a cylindrical can of total surface area A whose volume is as large as…

Q: Determine the volume if the area common with y^2 - 3x + 7y = 8 and x^2 - 4x +3 = y^2 is revolved…

A: Integration of function under certain limits with respect to any of the axis gives the area under…

Q: Show that the volume generated by revolving the region in the first quadrant bounded by the…

A: Solution is below.

Q: Show that a closed right circular cylinder of given surface has maximum volume if its height equals…

A: Let, the surface area be S, radius be r, height be h and volume be V. The total surface area be…

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 solid bounded by

A: Given,        z=-6+x2+y2,   and    z=6-2x2-2y2

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

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Q: Find the volume and the surface area of a “closed boOx' that has dimensions of 15 cm, 20 cm, and…

A: Given that: The dimensions of a " closed box" are 15 cm , 20 cm and 0.25m. 1m = 100 cm

Q: The cube is 3cm on one edge. Each of these joints A and B is 1cm from the vertex nearest it. If two…

A: This question is based on 3-D volume calculation in whivh two plane cut a wedge from a cube. We got…

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: Find the volume of the and the coordinate planes. tetrahedron bounded by 2x-y +z = 4

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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: Find the volume of the largest right circular cone that can be inserted in a sphere of radius 3.

A: The volume of the cone is given by  , where is the radius of the circular base of the cone and h is…

Q: Let m be the least possible surface area of a closed circular cylindrical can whose volume is equal…

A: • To solve this problem, we have to express the total surface area of the cylindrical can in a…

Q: Find the volume enclosed by the cy linder (x-1) ty²=1 and the sphere x² +y²7z²=L4.

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

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Q: 2. (2 - x² – y²)dydx

A: Given, I=∫01∫012-x2-y2dydx…

Q: Consider a rectangular box B that has a bottom and sides but no top and has minimal surface area…

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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 region that lies inside the sphere x2 + y2 + z2 = 2 and outside the cylinder…

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Q: Find the volume of c such that the area of the region bounded by the parabolas y = 16x2 - c2 and y =…

A: Given parabolas are y1=16x2-c2y2=c2-16x2 The area between the given parabolas is 1283

Q: Use the Theorem of Pappus to find the volume of the solid that is generated when the region enclosed…

A: y = 8−x² given y = x²     given This means that 8- x² = x² 2x2=8 x=-2,+2 Then, y=4,4

Q: Find the volume of the region cut from the solid sphere ρ≤ a by the half-planes θ = 0 and θ = π/6 in…

A: Find the volume of the region cut from the solid sphere.

Q: Find the volume of the region cut from the solid sphere

A: Given the region cut from the solid sphere x2+y2+z2≤4a2by the cylinder x2+y2=a2.

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: Find the volume enclosed by the torusp = 4 sin().-

A: The volume is given by: V=∫∫∫dv In spherical co-ordinate dV=ρ2sin φ dρ dφ dθ

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: 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: Calculate the volume of solid Q limited by surface z

A: Consider the solid Q: Clearly, 0≤z≤11+x2+y2Let x= r cos θ       y=r sin θ⇒x2+y2=r2Projection on the…

Q: A right cylinder of radius r is inscribed in a hollow sphere of fixed radius a. The

A: Given, a right cylinder of radius 'r', is inscribed in a hollow sphere of fixed radius 'a'. The…

Q: Find the volume.

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Q: = 1, 三 a NIU + +

A: Here Let u = x/a, v = y/b , w = z/c Then dx = a du, dy = b dv, dz = c dw So, Required volume V =…

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: Find the volume of the portion of the solid sphere ps8 that lies between the cones o= and ф %3D 3 3…

A: The portion of the solid sphere ρ≤8 that lies in between the cones ϕ=π3  and  ϕ=2π3

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

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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: Right circular cones of height h and radius r are attached to each end of a right circular cylinder…

A: Let Vcr be the volume of the cone and Vc be the volume of the cylinder. So, Vcr=πr2h and Vc=13πr2h.

Q: Find the dimensions of the closed right circular cylindrical can of smallest surface area whose…

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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…