Integrals Involving Roots — Question 6

Question 6 – Solution

Step 1: Evaluate the definite integral. $$\begin{align*} \int_0^a\sqrt{x}\,dx &=\int_0^a x^{1/2}\,dx\\ &=\left[\frac{x^{3/2}}{3/2}\right]_0^a\\ &=\left[\frac23x^{3/2}\right]_0^a =\frac23a^{3/2}. \end{align*}$$ Step 2: Set the result equal to 18 and solve. $$\begin{align*} \frac23a^{3/2}&=18,\\ a^{3/2}&=27,\\ \left(a^{3/2}\right)^{2/3}&=27^{2/3},\\ a&=(\sqrt[3]{27})^2=3^2=9. \end{align*}$$ $$\boxed{a=9}$$