### Home > APCALC > Chapter 2 > Lesson 2.3.2 > Problem2-117

2-117.

If $h(x) = \frac { 4 } { 5 }x^3 − 2x + 5$ use sigma notation to write Riemann sums to approximate the area under the curve for $10\le x\le15$ using $10, 20,$ and $100$ left endpoint rectangles of equal width. Then, use your calculator to determine these approximations. What happens as the number of rectangles increases?

$10$ left-endpoint rectangles:

$\displaystyle \sum_{t=0}^{9}\frac{1}{2}h\left ( 10+\frac{i}{2} \right )=\sum_{t=0}^{9}\frac{1}{2}\left ( \frac{4}{3}\left (10+\frac{i}{2} \right )^{3}-2\left ( 10+\frac{i}{2} \right )+5 \right )$

$20$ left-endpoint rectangles:

$100$ left-endpoint rectangles: