Basic Examples on Definite Integrals Set 1

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Example -1

Find the area under the curve \(\begin{align}y = \frac{1}{{{x^2} + 1}}\end{align}\) on its entire domain, i.e., evaluate  \(\begin{align}\int\limits_{ - \infty }^{ + \infty } {\frac{1}{{{x^2} + 1}}} \,\,dx\end{align}\)

Solution: The graph for \(\begin{align}y = \frac{1}{{{x^2} + 1}}\end{align}\) is sketched below:

Although the graph extends to infinity on both sides, the area under the curve will still be finite, as we’ll now see:

\[\begin{align}&\int\limits_{ - \infty }^{ + \infty } {\frac{1}{{{x^2} + 1}}} \,\,\,dx\,\, = \,\,\,\left. {{{\tan }^{ - 1}}x} \right|_{ - \infty }^{ + \infty }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\qquad\qquad\qquad\quad\;={\tan ^{ - 1}}\left( { + \infty } \right) - {\tan ^{ - 1}}\left( { - \infty } \right) \end{align}\]

What do we make of \({\tan ^{ - 1}}\left( { + \infty } \right)?\) We can take it to mean \(\mathop {\lim }\limits_{k \to \infty } \,\,{\tan ^{ - 1}}\left( k \right)\) which is \(\frac{\pi }{2}.\) Similarly, \({\tan ^{ - 1}}\left( { - \infty } \right)\) would equal \(\frac{{ - \pi }}{2}.\) Thus, the required area is \(\pi \) .

Example -2

Evaluate the area bounded between \(y = x\,\,{\rm{and}}\,\,y = {x^3}\) from x = 0 to x = 1.

Solution: The given curves are sketched in the region of interest below.

The required area is

\[\begin{align}&A = \int\limits_0^1 {\left( {x - {x^3}} \right)dx} \\ \,\,\,\, &\quad= \left. {\left( {\frac{{{x^2}}}{2} - \frac{{{x^4}}}{2}} \right)} \right|_0^1\\\,\,\, &\quad= \left( {\frac{1}{2} - \frac{1}{4}} \right) - \left( {0 - 0} \right)\\\,\,\, &\quad= \frac{1}{4}\end{align}\]

Example -3

Find the mean value of \(f\left( x \right) = {\cos ^2}x\,\,{\rm{on}}\left[ {0,\frac{\pi }{2}} \right]\)

Solution: Let \({f_{av}}\) be the required mean value.

\[\begin{align}&\Rightarrow\quad  \,{f_{av}}\left( {\frac{\pi }{2} - 0} \right) = \int\limits_0^{\pi /2} {{{\cos }^2}x\,dx} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\qquad\qquad\qquad\qquad= \frac{1}{2}\,\int\limits_0^{\pi /2} {\left( {1 + \cos 2x} \right)dx} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\qquad\qquad\qquad\qquad= \frac{1}{2}\left. {\left( {x + \frac{{\sin 2x}}{2}} \right)} \right|_0^{\pi /2}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\qquad\qquad\qquad\qquad= \frac{\pi }{4}\\&\qquad\qquad\Rightarrow \quad {f_{av}} = \frac{1}{2}\end{align}\]