The base of a loudspeaker is determined by the two curves $y=\dfrac{x^2}{10}$ and $y=-\dfrac{x^2}{10}$ for $1 \leq x \leq 4$, as shown in the figure above. For this loudspeaker, the cross sections perpendicular to the $x$-axis are squares. What is the volume of the loudspeaker, in cubic units?

Summary Submit

Each square has the area:

$$A = s^2$$ $$A = \left(\frac{x^2}{10}-\Big(-\frac{x^2}{10}\Big)\right)^2$$ $$A = \left( \frac{2x^2}{10}\right)^2$$ $$A = \frac{x^4}{25}$$

The volume can be calculated with the integral:

$$V=\int\limits_1^4 A   dx$$ $$= \int\limits_1^4 \frac{x^4}{25}   dx \approx \boxed{8.184}$$