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Volume of Revolution

Integration extends to the third dimension with the means to compute the volumes of shapes obtained by revolving regions around an axis, such as spheres, toroids, and paraboloids.

Problem Solving

If the surface area of the hypersphere (sphere in 4 dimensions) is \( 2 \pi ^2 R^3 \), what would be the volume of the hypersphere?

Consider a three dimensional object whose base is a semicircle with diameter \(AB = 28\). If each cross section of the three dimensional object cut by a plane perpendicular to \(\overline{AB}\) is a semicircle, as shown in the diagram above, what is the volume of the three dimensional object?

Consider a three-dimensional shape standing on the floor with height \(\frac{\pi}{2}\text{ cm}\) such that the area of a cross section parallel to the floor is \(14\sin x\cdot \cos x \text{ cm}^2\) for any height \(0 \le x \le \frac{\pi}{2}\text{ cm}.\) What is the volume of the three-dimensional shape in \(\text{cm}^3\)?

Consider a three-dimensional shape standing on the floor with height \(17 \text{ cm}\) such that the area of a cross section parallel to the floor is \( \frac{3}{2} \sqrt{17-x} \text{ cm}^2\) for any height \(0 \le x \le 17 .\) What is the volume of the three-dimensional shape in \(\text{cm}^3\)?

In the above diagram, the equation of the curve is \(y=\sqrt{ax}=\sqrt{2x}\) and the equation of the line is \(y=mx.\) Let \(V_x\) and \(V_y\) be the volumes of the solids obtained by revolving the shaded region bounded by the curve and line about the \(x\)-axis and \(y\)-axis, respectively. If \(V_x=V_y,\) what is the value of \(m?\)

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