Volumes with Cross Sections: Triangles and Semicircles AB & BC

Only the constant changes from 8.7. An equilateral triangle of side $s$ has height $\frac{\sqrt{3}}{2}s$ and area $\frac{\sqrt{3}}{4}s^{2}$, so keeping the height as the area doubles the answer. An isosceles right triangle has area $\frac{1}{2}s^{2}$ when the base is a leg and $\frac{s^{2}}{4}$ when the base is the hypotenuse, since the altitude to a hypotenuse of length $s$ is $\frac{s}{2}$. The phrase in the problem is what separates the two, and the factor between them is $2$.

A semicircle standing on a base has that base as its diameter, so $r = \frac{s}{2}$ and $A = \frac{1}{2}\pi r^{2} = \frac{\pi}{8}s^{2}$; using $s$ as the radius is four times too large, and forgetting to halve the circle is twice too large. Writing $r = \frac{s}{2}$ on its own line before substituting prevents both. Over the base under $y = \sqrt{x}$ on $[0, 4]$, where $s^{2} = x$ and $\int_0^4 x\,dx = 8$, the three volumes are $2\sqrt{3}$, $4$ and $\pi$, and the right triangle's solid being the largest and the semicircle's the smallest is a check on the constants.