Left Boxes

The sum of the areas enclosed by rectangles is the Riemann sum

$\displaystyle {\frac{{b-a}}{{n}}}$$\displaystyle \sum_{{i=0}}^{{n-1}}$f$\displaystyle \left(\vphantom{ a+i\frac{b-a}{n}}\right.$a + i$\displaystyle {\frac{{b-a}}{{n}}}$$\displaystyle \left.\vphantom{ a+i\frac{b-a}{n}}\right)$

where the heights of the rectangles are determined by the function values at the left-hand endpoints of the subintervals.

$\blacktriangleright$ To make a left-boxes plot

1.
Make a middle-boxes plot.

2.
Select the frame or the view.

3.
Click itbpF0.3009in0.3009in0.0701inproperty.wmfand choose the Plot Components page.

4.
Check Left Boxes.

5.
Reset the intervals and the number of boxes as desired.

6.
Choose OK.

$\blacktriangleright$ Calculus + Plot Approx. Integral, Edit + Properties

x sin x

dtbpF3in2.0003in0pt

Applied to the expression x sin x, with four rectangles and 0≤x≤3, the approximating Riemann sum is

$\displaystyle {\frac{{3}}{{4}}}$$\displaystyle \sum_{{i=0}}^{{3}}$$\displaystyle \left(\vphantom{ i\frac{3}{4}}\right.$i$\displaystyle {\frac{{3}}{{4}}}$$\displaystyle \left.\vphantom{ i\frac{3}{4}}\right)$sin$\displaystyle \left(\vphantom{ i\frac{3}{4}%
}\right.$i$\displaystyle {\frac{{3}}{{4}%
}}$$\displaystyle \left.\vphantom{ i\frac{3}{4}%
}\right)$ = 2.818602182