Question

Graph the function \(f(x) = {\cos ^2}x\sin x\) and use the graph to guess the value of the integral \(\int_0^{2\pi } {f(x)dx} \). Then evaluate the integral to confirm your guess

Short answer

The value for the integral \(f(x) = {\cos ^2}x\sin x\) using the graph and then evaluating confirms that it results in \(0\).

Step-by-step solution

Integral definition

In mathematics, an integral is a numerical number equal to the area under a function's graph for some interval or a new function whose derivative equals the original function

Plotting the graph

The graph for the function \(f(x) = {\cos ^2}x\sin x\)is plotted below –

All the areas above the \(x - \)axis will be cancelled out by the areas below the \(x - \)axis.

Simplifying the function

Evaluating such that –

\(\begin{aligned}{c}u &= \cos x\\du &= - \sin xdx\\ - du &= \sin xdx\end{aligned}\)

Now we set the new limits of integration for\(u\)-

\(u = \left\{ {\begin{aligned}{*{20}{l}}{1,}&{{\rm{ if }}x = 2\pi }\\{1,}&{{\rm{ if }}x = 0}\end{aligned}} \right.\)

Simplifying the function \(\int_0^{2\pi } {{{\cos }^2}} (x){\sin ^3}(x)dx\) –

\(\begin{aligned}{c}\int_0^{2\pi } {{{\cos }^2}} (x){\sin ^3}(x)dx &= \int_0^{2\pi } {{{\cos }^2}} (x){\sin ^2}(x)\sin (x)dx\\ &= \int_0^{2\pi } {{{\cos }^2}} (x)\left( {1 - {{\cos }^2}(x)} \right)\sin (x)dx\end{aligned}\)

Substituting the values

Substituting u to function\(\int_0^{2\pi } {{{\cos }^2}} (x)\left( {1 - {{\cos }^2}(x)} \right)\sin (x)dx\)–

\(\begin{aligned}{c}\int_0^{2\pi } {{{\cos }^2}} (x)\left( {1 - {{\cos }^2}(x)} \right)\sin (x)dx &= \int_1^1 {{u^2}} \left( {1 - {u^2}} \right)( - du)\\ &= - \int_1^1 {{u^2}} \left( {1 - {u^2}} \right)du\end{aligned}\)

Using the properties of Definite Integral, if lower bound is the same with the upper bound, then the change in \(x\) is equal to \(0\).

\(\begin{aligned}{c}\int_0^{2\pi } {{{\cos }^2}} (x)\left( {1 - {{\cos }^2}(x)} \right)\sin (x)dx &= - \int_1^1 {{u^2}} \left( {1 - {u^2}} \right)du\\ &= 0\end{aligned}\)

Therefore, using the graph and the evaluating the value is obtained as \(0\).