Calculus
# Volume of Revolution

$2 \pi ^2 R^3$, what would be the volume of the hypersphere?

If the surface area of the hypersphere (sphere in 4 dimensions) is$AB = 28$. 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 object whose base is a semicircle with diameter$y=\sqrt{ax}=\sqrt{2x}$ 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 $m?$

In the above diagram, the equation of the curve is