Blog: `y = x/(x^2 + 1)^2,` `y = x^5 - x,` `xgt=0` Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then...

Saturday, September 10, 2016

We can see that these graphs intersect at x = 0, and around x = 1. A little greater than x = 1.

In order to approximate the area of this region we take the integral of the top function minus the bottom function from x = 0 to x = 1.

$\displaystyle{\int_{{0}}^{{1}}}\frac{{x}}{{{\left({{x}}^{{2}}+{1}\right)}}^{{2}}}-{\left({{x}}^{{5}}-{x}\right)}{\left.{d}{x}\right.}$

We can use u sub to find the integral of the first part, which will equal

$\displaystyle-\frac{{1}}{{{2}{\left({{x}}^{{2}}+{1}\right)}}}$

And the second part is just a simple power rule. So we get:

$\displaystyle-\frac{{1}}{{{2}{\left({{x}}^{{2}}+{1}\right)}}}-\frac{{{x}}^{{6}}}{{6}}-\frac{{{x}}^{{2}}}{{2}}$ from 0 to 1.

When evaluated gets us 7/12, or about .5833

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