The base of a solid is a region in the first quadrant bounded by the $x$-axis, the $y$-axis, and the line $x+2y=8$, as shown in the figure above. If cross sections of the solid perpendicular to the $x$-axis are semicircles, what is the volume of the solid?

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Each semicircle has a diameter equal to the $y$-value of the line. For example, the semicircle at the origin has a diameter of $4$, corresponding to the point on the line $(0,4)$.

The area of each semicircle is:

$$A = \frac{1}{2}\pi r^2$$

The radius is half of the $y$-coordinate/diameter.

$$x+2y=8$$ $$y=-\frac{1}{2}x+4$$ $$r=\frac{y}{2}$$ $$r = -\frac{1}{4}x+2$$

The volume is given by the following integral:

$$V = \int\limits_0^8 A_{\text{semicircle}}   dx$$ $$V = \int\limits_0^8 \frac{1}{2}\pi \left(-\frac{1}{4}x+2\right)^2   dx$$ $$V = \frac{1}{2}\pi \int\limits_0^8 \left(-\frac{1}{4}x+2\right)^2   dx$$

Solve this definite integral using your calculator or by hand.

$$V = \frac{1}{2}\pi \int\limits_0^8 \left(-\frac{1}{4}x+2\right)^2   dx$$ $$V = \frac{1}{2}\pi \left(\frac{32}{3}\right)$$ $$V \approx \boxed{16.755}$$