Answer
$\frac{\pi}{3}b^3$
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Work Step by Step
Step 1. Divide the interval $[0,b]$ into $n$ equal parts with width $\Delta x=\frac{b}{n}$.
Step 2. For the $k$th part, $x_k=k\Delta x, y_k=\pi x_k^2=\pi k^2(\Delta x)^2$; the area for this part is $\Delta A_k=y_k\Delta x=\pi k^2(\Delta x)^3=\pi k^2(\frac{b}{n})^3=\frac{\pi k^2b^3}{n^3}$
Step 3. Add up the area of all the rectangles: $A=\Sigma^n_{k=1}A_k=\Sigma^n_{k=1}\frac{\pi k^2b^3}{n^3}=\frac{\pi b^3}{n^3}\Sigma^n_{k=1}k^2=\frac{\pi b^3}{n^3}\times\frac{n(n+1)(2n+1)}{6}=\frac{\pi b^3(1+1/n)(2+1/n)}{6}$
Step 4. The area of the region is then given by $\lim_{n\to\infty}A=\lim_{n\to\infty}\frac{\pi b^3(1+1/n)(2+1/n)}{6}=\frac{\pi}{3}b^3$