Notes: `y = xsin(x^2), y = x^4` Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find...

Friday, 13 March 2015

$\displaystyle{y}={x}{\sin{{\left({{x}}^{{2}}\right)}}},{y}={{x}}^{{4}}$ Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find...

The red curve refers to the graph of the first function: $\displaystyle{y}={x}{\sin{{\left({{x}}^{{2}}\right)}}}$ while the the the blue curve refers to the graph of the second function: $\displaystyle{y}={{x}}^{{4}}$ .

As shown in the xy-plane, the two graphs intersect , approximately, at the following points: (0,0) and (0.9,0.65).

Based on these intersection points, the limits of integration with respect to x will be from x= 0 to x=0.9.

The formula for the " Area...

As shown in the xy-plane, the two graphs intersect , approximately, at the following points: (0,0) and (0.9,0.65).

Based on these intersection points, the limits of integration with respect to x will be from x= 0 to x=0.9.

The formula for the " Area between Two Curves" is:

A= $\displaystyle{\int_{{a}}^{{b}}}{\left[{f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}\right]}{\left.{d}{x}\right.}$

such that $\displaystyle{f{{\left({x}\right)}}}\geq{g{{\left({x}\right)}}}$ on the interval of [a,b].

This is the same as A = $\displaystyle{\int_{{a}}^{{b}}}{\left[{y}_{{{a}{b}{o}{v}{e}}}-{y}_{{{b}{e}{l}{o}{w}}}\right]}{\left.{d}{x}\right.}$

where the bounded area is in between $\displaystyle{y}_{{{a}{b}{o}{v}{e}}}={f{{\left({x}\right)}}}$ and $\displaystyle{y}_{{{b}{e}{l}{o}{w}}}={g{{\left({x}\right)}}}$ .

Applying the formula on the given problem, the integration will be:

A = $\displaystyle{\int_{{0}}^{{{0.9}}}}{\left[{x}\cdot{\sin{{\left({{x}}^{{2}}\right)}}}-{{x}}^{{4}}\right]}{\left.{d}{x}\right.}$

= $\displaystyle{\left[-{\left(\frac{{\cos{{\left({{x}}^{{2}}\right)}}}}{{2}}\right)}-\frac{{{x}}^{{5}}}{{5}}\right]}{{\mid}_{{0}}^{{{0.9}}}}$

= $\displaystyle{\left[-{\left(\frac{{\cos{{\left({{\left({0.9}\right)}}^{{2}}\right)}}}}{{2}}\right)}-\frac{{{\left({0.9}\right)}}^{{5}}}{{5}}\right]}-{\left[-{\left(\frac{{\cos{{\left({{\left({0}\right)}}^{{2}}\right)}}}}{{2}}\right)}-\frac{{{0}}^{{5}}}{{5}}\right]}$