More Volume Problems — Question 2

Question 2 – Solution

1. Choose slices and bounds.

The cross-sections are perpendicular to the $x$-axis, so use slices of thickness $dx$ on the given interval $0\le x\le4$.

2. Find the square side length.

The vertical segment in the base is the side of the square:$$s(x)=y_{\text{top}}-y_{\text{bottom}}=\sqrt{x}-0=\sqrt{x}.$$

3. Write the cross-sectional area.

For a square, area is side squared:$$A(x)=s(x)^2=(\sqrt{x})^2=x.$$

4. Set up and integrate the volume.

Add the areas of the thin cross-sections:$$V=\int_0^4A(x)\,dx=\int_0^4 x\,dx=\left[\frac{x^2}{2}\right]_0^4.$$

5. Evaluate both bounds.

$$V=\frac{4^2}{2}-\frac{0^2}{2}=\frac{16}{2}-0=\boxed{8}.$$

All volumes are in cubic units.