Right Boxes

For right boxes , the sum of the areas enclosed by rectangles is the Riemann sum

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

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

$\blacktriangleright$ To make a right-boxes plot

$\blacktriangleright$ Calculus + Plot Approx. Integral

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}}}$$\displaystyle \sum_{{i=1}}^{{4}}$$\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)$ = 3.1361222