Question

Do case (b) of Example $\mathbf{1}\mathbf{}$above.

Short answer

The value of the integral is $\mathrm{I}=-3{\mathrm{\pi a}}^2$.

Step-by-step solution

Given Information

The movement in the counter clockwise direction on the $\mathrm{XY}$plane is $\mathrm{dr}=-\mathrm{dx}\widehat{\mathrm{i}}+\mathrm{dy}\widehat{\mathrm{j}}$.

Definition of Integration

In mathematics, integration is a process of finding a function$\mathbf{g}\mathbf{\left(}\mathbf{X}\mathbf{\right)}$ whose derivative, $\mathbf{Dg}\mathbf{\left(}\mathbf{X}\mathbf{\right)}$, is equal to a given function $\mathit{f}\mathbf{\left(}\mathbf{X}\mathbf{\right)}$.

Use the curve equation and integrate

The movement in the counter clockwise direction on the XY plane is $\mathrm{dr}=-\mathrm{dx}\widehat{\mathrm{i}}+\mathrm{dy}\widehat{\mathrm{j}}$.

So, $\mathrm{Vdr}=-4\mathrm{ydx}+\mathrm{xdy}$.

The equation of the curve is ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2$

This implies:

$\mathrm{y}=\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}$

$\mathrm{dy}=\frac{-\mathrm{x}}{\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}$

The value of the integral is:

$\mathrm{I}=4{\int }_0^{\mathrm{a}} \left[-\mathrm{a}\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}+\frac{{\mathrm{x}}^2}{\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}\right]\mathrm{dx}$

$=4{\int }_0^{\mathrm{a}} \frac{5{\mathrm{x}}^2-4{\mathrm{a}}^2}{\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}\mathrm{dx}$

Change the variables.

$\mathrm{x}=\mathrm{asin}\mathrm{\theta }$

$\mathrm{dx}=\mathrm{acos}\mathrm{\theta d\theta }$

Change the limit.

$0\to 0$and $\mathrm{a}\to \frac{\mathrm{\pi }}{2}$

Hence, the value of the integral is $\mathrm{I}=-3{\mathrm{\pi a}}^2$