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

Q: Find the volume of the 'cover' that is bounded by 52 < x2 + y2 + z² < 7² and the upper cone z² = 4x2…

A: Given 52≤x2+y2+z2≤72 ,z2=4x2+4y2

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 generated by revolving about the line y-2=0, the area bounded by x^2=8y and the line…

A: By using cylindrical shell &amp; horizontal strip (Shell Method);The Volume of solid formed by…

Q: Determine the dimensions of the right circular cylinder of maximum lateral surface area that can be…

A: Maximum lateral surface area will occur at critical point  So put ds/d(theta) = 0 Find theta

Q: Find the volume of the region under the surface 2 and above the triangle in the xy-plane = Z 1+ x2…

A: Given,z=41+x2triangles co-ordinates :  (0,0,0),(8,0,0) and (8,8,0)On XY planetherefore Limits:…

Q: Find the volume of the solid that is bounded by the surfaces z = = 92 - x2 - y2 in the first octant.…

A: 0≤ z ≤ 81-x2-y2z=0x2+y2=920≤y≤81-x20≤x≤9

Q: A right circular cylinder is inscribed in a sphere of radius r .Find the largest possible volume of…

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Q: The area bounded by the lines x = 0, x = 2, x + y = 4, and x – y = 4 is revolved about  the line x =…

A: Given problem:- The area bounded by the lines x = 0, x = 2, x + y = 4, and x – y = 4 is revolved…

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: ind the center of mass for a thin plate covering the region bounded by the parabola y=25-x^2 and…

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Q: 2. bounded by the axis of the region Volume of the body of rotation created by rotating it Oround ?

A: Given that  y = 1 -x2, x =0 , y = 0   x2 = 1- y The limits of y :-  from y = 0 to y = 1

Q: A right circular cylinder is inscribed in a sphere of radius r. Find the largest possiblesurface…

A: Consider the right circular cylinder inscribed in the sphere with radius, r. Suppose the height and…

Q: Determine the right circular cone of maximum volume that can be inscribed inside a given sphere of…

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Q: Let S be the intersection of two cylinders of radius r whose axes intersect at an angle θ. Find the…

A: Each cross-section at a distance y from the central axis(for -r≤y≤r) is a rhombus with side length…

Q: A right circular cone is circumscribed by a sphere of radius 1. Determine the height h and radius r…

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Q: Find the volume of the 'cover' that is bounded by 32 < x2 + y2 + z2 < 72 and the upper cone z2 = 5x2…

A: To find the volume of the solid that is bounded by 32≤x2+y2+z2≤72 and the upper cone z2=5x2+5y2.

Q: Find the volume of the region under the surface z and above the triangle in the xy-plane with 1+ x²…

A: We have to find the volume of the region under the surface.

Q: Find theirelume of the largest. right circular Cone that can be inscribed in a sphere of radius 3.

A: we know,     The volume  of right circular cone is:      V =1/3 pi*r2*h  , where r is the radius of…

Q: The base of a three-dimensional figure is bound by the line y = 4 sin(x) on the %3D interval [0, 1].…

A: Consider a vertical cross-section of the required solid with thickness, as shown below: The volume…

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 center of mass of a thin plate of constant density & = 2 covering the given region The…

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Q: Determine the volume of the solid that will be generated by revolving the area bounded by the given…

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Q: 3. Show that a closed right circular cylinder of given surface has maximum volume if its height…

A: Let r, h be the radius and height of the cylinder respectively and V, S be the Volume and Surface…

Q: Y = x x = 0 y = 8 is revolved about the y-axis

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Q: Find the volume generated by revolvingg the area bounded by the given curve and awd lines y = x', x…

A: Volume generated by revolving the area bounded by curve  y=x3  and line x=0  and y=8 (that is 8=x3…

Q: Let R be the intersection of the circles of radius 1 centered at (1, 0) and (0, 1). Express as an…

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Q: Find the volume of the region under the surface z = and above the triangle in the xy-plane with 1+…

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Q: Show that the volume of the largest right circular cylinder that 4 the volume 9. can be inscribed in…

A: The largest circular cylinder that can be inscribed in a given right circular cone is 4/9 the volume…

Q: Find the volume generated by revolving bout the line x = -1 the area bounded by (y + 2)? = 4(x + 2)…

A: Let R be the reion bounded by y+22=4x+2 and x=-1.We have to find the volume generated by revolving…

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

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

Q: Find the volume generated by revolving bout the line x -1 the area bounded by (y + 2) = 4(x + 2) and…

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Q: Find the volume generated by revolving bout the line x = 3 the area bounded by (y - 2)* = -4(x - 4)…

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Q: Calculate the volume of the region bounded by the spherex + y² + z² = 9 from above and by the cone…

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Q: Use spherical coordinates to find the volume of a sphere of radius 2 from which a central cy tinder…

A: Let us assume that , x2+y2+z2=22x2+y2=1 The graph will be :

Q: 2. Find the volume when the plane area bounded by y = -x² – 3x + 6 and x + 3 = 0 is revolved (a)…

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Q: FIND THE VOLUME GENERATED BY REVOLVING THE AREA BOUNDED BY THE GIVEN CURVES AND LINES REVOLVED ABOUT…

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Q: Find the volume of the solid that is bounded by the xz-plane and the hemispheres y=sqrt(9-x^2-z^2)…

A: Since it is a hemisphere between two spheres of radius 3 and 4 so the volume is:  23π43-33=23π64-27…

Q: Find the volume generated by revolving about the y-axis in the first quadrant bounded by y^2-3x+3=0…

A: We have, y2-3x+3=0⇒x=y2+33 x-4=0   Volume generated by revolving the bounded region can be…

Q: An oil storage tank can be described as the volume generated by revolving the area bounded by…

A: Given: y=49+x214, x=0, y=0, x=2 To determine: The volume of the tank generated by revolving the area…

Q: By use of cylindrical shell method, find the volume generated by revolving the area bounded by y =…

A: The volume generated by revolving the area bounded by a curve y=fx and y=gx about y-axis using…

Q: he volume bounded by y = x², y = 0 and x = 2 rotated about the specified = specified orientation of…

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Q: Find the volume of the solid that is bounded by the surfaces z = 92 - x2 - y2 in the first octant.…

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Q: find volume of area bounded by y = x 2x +3 and y = 3 about line y = 3 an |the

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Q: Find the volume generated by revolving bout the line x = -1 the area bounded by (y + 2)° = 4(x + 2)…

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Q: Find the Volume generated by revolving the area bounded by the given curves and lines revolved about…

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Q: Find the volumes of the solids generated by revolving the regions bounded by the graphs of the…

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Q: 4. Find the volume generated(disk) if the area bounded by the 3 curves y = 1 – y = 0, x = 1 and…

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Cylinders

A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.

Cones

A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.