64. if (f) is continuous and (int_{0}^{9}f(x)dx = 4), find (int_{0}^{3}xf(x^{2})dx).

Explanation:

Step1: Use substitution

Let $u = x^{2}$, then $du=2x dx$. When $x = 0$, $u = 0$; when $x = 3$, $u=9$. And $x dx=\frac{1}{2}du$.

Step2: Rewrite the integral

$\int_{0}^{3}xf(x^{2})dx=\frac{1}{2}\int_{0}^{9}f(u)du$.

Step3: Substitute the given value

Since $\int_{0}^{9}f(x)dx = 4$, then $\frac{1}{2}\int_{0}^{9}f(u)du=\frac{1}{2}\times4$.

Answer:

$2$