A torus is generated by rotating the circle x2 + (y – R)² = r² about the x-axis. Find the volume enclosed by the torus.

Q: Find the volume of the cap of a sphere of radius r and height h for h < r.

A: The equation of the sphere with radius r is x2+y2+z2=r2. Determine z2 from the equation of sphere.…

Q: Find the volume of the solid S whose base is the disk bounded by the circle x^2 + y^2 = r^2 and the…

A: Solution is given below:

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

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Q: ind the valume of the portion of the Solid. Sphere x+y2,+2-10 that les between the Surfaces

A: We need to find the volume of the portion of the solid sphere x2+y2+z2 = 10 that lies between the…

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

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Q: Use the disk method to verify that the volume of a sphere is 4/3 πr3 , where r is the radius ?

A: So sphere is basically a circle that is rotated through the x-axis. We will suppose to cut a disk…

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

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

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Q: Use the Theorem of Pappus to find the exact volume of a general cone c with vertices (0, 0), (3, 0),…

A: Step:-1 Pappu's Theorem:- According to this theorem, the Volume V of a solid of revolution generated…

Q: Calculate the volume generated by the circle x^2 + y^2 -4x=-3 that revolves around ox

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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 of the solid that is enclosed by the cone  z = sqrt(x2 + y2) and the sphere  x2 +…

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Q: Find the volume of the pyramid with base in the plane z = -1 and sides formed by the three planes y…

A: We need to evaluate volume.

Q: Find the volume of the solid that lies under the elliptical paraboloid z = 2x? + y + 4 and above the…

A: V=∫∫RzdA V=∫01∫012x2+y2+4dydx Integrate with respect to y, V=∫012x2y+y33+y01dx…

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: Find the volume of the region enclosed by the paraboloid (x-1)²+y=z and the plane 2x+z=2.

A: To find the volume of the region enclosed by below paraboloid and the plane. x−12+y2=z and…

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

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Q: Use the Theorem of Pappus and the fact that the volume ofa sphere of radius r  is 4/3πr3  to find…

A: We have, The volume of a sphere is V=43πr3                               ⋯⋯1. The semi-circular…

Q: The volume of the tetrahedron enclosed by the coordinate planes and the plane 9x +y +Z = 18 is ?

A: We want to find the volume of the tetrahedron enclosed by the co-ordinate planes, i.e. x=0, y=0, z=0…

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

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

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Q: Find the volume of the solid that is enclosed by the cone z = x² + y² and the sphere x² + y² + z² =…

A: Given: Equation of cone z=x2+y2 Equation of sphere x2+y2+z2=2

Q: Find the volume cm^3 of the solid figure in Fig 1182 that was generated by rotating the shaded…

A: Volume of revolution

Q: Use the disk method to verify that the volume of a right circular cone is 1/3 πr2 h, where r is the…

A: Given that: verify that the volume of a right circular cone is 1/3 πr2 h, where r is the radius of…

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

A:

Q: Let a sphere of radius r be cut by a plane, thereby forming a segment of height h. Show that the…

A: Sphere was cut at height h, radius of the circle at intersection of 2 cuts is r2-h2. Similarly…

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: use the disk method to find the volume (in units^3) when the regionbounded by the curves x+y=5, x=0,…

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Q: The volume of the parallelepiped is Volume = |a x b||c|| sin | = |(a × %3D b). c| True False

A: Using direct formula

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 that lies under the elliptical paraboloid z = 2x2 + y? +4 and above the…

A: Given, z=2x2+y2+4 To find Volume of Solid

Q: Find the surface area of a sphere of radius r

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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: find the surface area obtained by rotating the circle (x – 4)2 + y? =1 about the y-axis. %3D

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Q: (b) Prove the volume of a parallelepiped with a vertex at the origin and edges ü,v, and w is given…

A: In the question we have to prove the formula.

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

A: the circle: (x − 7)2 + y2 = 16 = 42 Compare it with standard equation of a circle: (x - a)2 + (y -…

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: Calculate the volume of the region bounded by the sphere a2 + y? + z? = 5 from above and by the cone…

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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: If we have a half-circle of radius r centered at 0, prove thatthe volume of the sphere resulting…

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Q: Let a sphere of radius r be cut by a plane, thereby forming a segment of height h. Show that the…

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Q: Find the volume of the solid that is enclosed by the cone z = and the sphere x² + y2 + z2 = 72.

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

A: We have to find the volume of the torus by using Pappus theorem.

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: Use the disk method to verify that the volume of a sphere is (4/3)π r3 , where r is the radius.

A: Volume of sphere of radius r is (4/3) πr3

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…