Cubic volume

1.Explain the term Cubic volume – “cubature”!

Solution:

Cubic volume – Cubature is the calculation of the volume of solids of revolution. We compute the volume of a body formed by rotating a plane figure (a rectangle, triangle, trapezoid, circle, etc.) about the x-axis. The plane figure is bounded by the x-axis, the lines x1 = a, x2 = b, and the curve y = f(x).

urcity-integral-kubatura-1 

2.Derive the formula for the volume of a right circular cylinder with base radius r and height v.

Solution:

The solid is formed by rotating a rectangle about the x-axis. The rectangle is bounded by the x-axis and the lines x1=a=0, x2=b= v, y = r.

urcity-integral-kubatura-2

3. Derive the formula for the volume of a right circular cone whose base radius is r and height is v.

Solution:

urcity-integral-kubatura-3 

4.Derive the formula for the volume of a sphere with radius r!

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5.Derive the formula for the volume of a rotational ellipsoid with semi-axes a, b.

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6.Derive the formula for the volume of a rotational paraboloid formed by rotating the parabola y2 = 2px about the x-axis. The paraboloid has height v.

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7.Compute the volume of the solid obtained by rotating the function y = cos x about the x-axis.

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8. Derive the formula for the volume of a frustum of a cone. The lower base has radius R, the upper base r. The height of the cone is v.

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