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.

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?

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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 \(\text{cm}^3\)?

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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 \(\text{cm}^3\)?

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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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