### Home > CALC > Chapter 2 > Lesson 2.2.2 > Problem2-69

2-69.

Using sigma notation, write a Riemann sum to estimate the area under the function $f(x) = x \text{ cos }x$ for $−2 ≤ x ≤ 6$ with eight left endpoint rectangles of equal width. Then use the summation feature of your graphing calculator to calculate the estimated area.

Since area of a rectangle $=$ (base)(height), start by finding the base, or $Δx$ of each rectangle. Divide the interval by the number of rectangles:

Next, find the height of each rectangle. The height is determined by $f(a +Δxi)$, where $a =$ starting value on the $x$-axis.
$f((−2) + (1)i) = f(−2 + i)$

Finally, use sigma notation to represent the sum of all eight rectangles. Since these are left-endpoint rectangles, the index will start at $0$ and end at $n − 1$, where $n =$ number of rectangles.

Use your calculator to evaluate this sum.