Integrals Involving Roots — Question 3

Question 3 – Solution

Step 1: Split at the $x$-intercept. The function is negative on $[-1,0]$ and positive on $[0,8]$, so $$A=-\int_{-1}^0x^{1/3}\,dx+\int_0^8x^{1/3}\,dx.$$ Step 2: Find an antiderivative. $$\int x^{1/3}\,dx=\frac{x^{4/3}}{4/3}=\frac34x^{4/3}.$$ Step 3: Evaluate both pieces. $$\begin{align*} A&=-\left[\frac34x^{4/3}\right]_{-1}^{0} +\left[\frac34x^{4/3}\right]_{0}^{8}\\ &=-\left(0-\frac34\right)+\left(\frac34\cdot16-0\right)\\ &=\frac34+12=\frac{51}{4}. \end{align*}$$ $$\boxed{A=\frac{51}{4}}$$