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:

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