Notes: `y = e^x, y = x(e^x), x = 0` Sketch the region enclosed by the given curves and find its area.

Wednesday, 8 February 2017

$\displaystyle{y}={{e}}^{{x}},{y}={x}{\left({{e}}^{{x}}\right)},{x}={0}$ Sketch the region enclosed by the given curves and find its area.

Graph each equation to determine the bounded region.

( Red curve is the graph of $\displaystyle{y}={{e}}^{{x}}$ . Blue curve is the graph of $\displaystyle{y}={x}{{e}}^{{x}}$ . And green line is the graph of x=0.)

Base on the graph, the bounded region of the three equations starts at x=0 and ends at x =1. Then, draw a vertical strip inside the region. Apply the formula:

$\displaystyle{A}={\int_{{a}}^{{b}}}{\left({y}_{{U}}-{y}_{{L}}\right)}{\left.{d}{x}\right.}$

(Refer to the attached figure below.)

The upper...

Graph each equation to determine the bounded region.

( Red curve is the graph of $\displaystyle{y}={{e}}^{{x}}$ . Blue curve is the graph of $\displaystyle{y}={x}{{e}}^{{x}}$ . And green line is the graph of x=0.)

Base on the graph, the bounded region of the three equations starts at x=0 and ends at x =1. Then, draw a vertical strip inside the region. Apply the formula:

$\displaystyle{A}={\int_{{a}}^{{b}}}{\left({y}_{{U}}-{y}_{{L}}\right)}{\left.{d}{x}\right.}$

(Refer to the attached figure below.)

The upper end of the vertical strip touches the curve $\displaystyle{y}={{e}}^{{x}}$ . And its lower end touches the curve $\displaystyle{y}={x}{{e}}^{{x}}$ . So the integral will be:

$\displaystyle{A}={\int_{{0}}^{{1}}}{\left({{e}}^{{x}}-{x}{{e}}^{{x}}\right)}{\left.{d}{x}\right.}$

$\displaystyle{A}={\int_{{0}}^{{1}}}{{e}}^{{x}}{\left({1}-{x}\right)}{\left.{d}{x}\right.}$

To take the integral of this, apply integration by parts $\displaystyle\int{u}{d}{v}={u}\cdot{v}-\int{v}{d}{u}$ .

In the integrand of area, u , du , v and dv are:

$\displaystyle{u}={1}-{x}$

$\displaystyle{d}{u}=-{\left.{d}{x}\right.}$

$\displaystyle{d}{v}={{e}}^{{x}}{\left.{d}{x}\right.}$

$\displaystyle{v}={{e}}^{{x}}$

Plugging them to the formula integration by parts, then A becomes:

$\displaystyle{A}={\left({1}-{x}\right)}\cdot{{e}}^{{x}}-\int{{e}}^{{x}}\cdot{\left(-{\left.{d}{x}\right.}\right)}={\left({1}-{x}\right)}{{e}}^{{x}}+\int{{e}}^{{x}}{\left.{d}{x}\right.}$

$\displaystyle{A}={\left({\left({1}-{x}\right)}{{e}}^{{x}}+{{e}}^{{x}}\right)}{{\mid}_{{0}}^{{1}}}={{\left({{e}}^{{x}}-{x}{{e}}^{{x}}+{{e}}^{{x}}\right)}_{{0}}^{{1}}}$