How do you Use a Riemann sum to find area?

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Answer 1 · 2014-10-10T18:38:35

The area A of the region under the graph of $f$ above the $x$ -axis from $x=a$ to $b$ can be found by

$A=lim_{n to infty}sum_{i=1}^n f(x_i) Delta x$ ,

where $x_i=a+iDelta x$ and $Delta x={b-a}/n$ .

Let us find the area of the region under the graph of $y=2x+1$ from $x=1$ to $3$ .

By definition,

$A=lim_{n to infty}sum_{i=1}^n[2(1+2/ni)+1]2/n$

by simplifying the expression inside the summation,

$=lim_{n to infty}sum_{i=1}^n(8/n^2i+6/n)$

by splitting the summation and pulling out constants,

$=lim_{n to infty}(8/n^2sum_{i=1}^ni+6/nsum_{i=1}^n1)$

by the summation formulas $sum_{i=1}^ni={n(n+1)}/2$ and $sum_{i=1}^n1=n$ ,

$=lim_{n to infty}(8/n^2cdot{n(n+1)}/2+6/ncdot n)$

by cancelling out $n$ 's,

$=lim_{n to infty}[4(1+1/n)+6]=4(1+0)+6=10$

I hope that this was helpful.