Answer
$b^2$
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, $x_k=k\Delta x, y_k=2x_k=2k(\frac{b}{n})$; the area for this part is $A_k=y_k\Delta x=2k(\frac{b}{n})^2=\frac{2b^2}{n^2}k$
Step 3. Add up the area of all the rectangles: $A=\Sigma^n_{k=1}A_k=\Sigma^n_{k=1}\frac{2b^2}{n^2}k=\frac{2b^2}{n^2}\Sigma^n_{k=1}k=\frac{2b^2}{n^2}\times\frac{n(n+1)}{2}=\frac{(n+1)b^2}{n}$
Step 4. The area of the region is then given by $\lim_{n\to\infty}A=\lim_{n\to\infty}\frac{(n+1)b^2}{n}=b^2$
