Question

\[\int\limits_0^9 f\left( x \right) dx, where f\left( x \right) \begin{cases}\sin x & , & 0 \leq x \leq \pi/2 \\ 1 & , & \pi/2 \leq x \leq 3 \\ e^{x - 3} & , & 3 \leq x \leq 9\end{cases}\]

Answer 1

We have,

\[\int\limits_0^9 f\left( x \right) dx, where f\left( x \right) \begin{cases}\sin x & , & 0 \leq x \leq \frac{\pi}{2} \\ 1 & , & \frac{\pi}{2} \leq x \leq 3 \\ e^{x - 3} & , & 3 \leq x \leq 9\end{cases}\]

\[I = \int_0^9 f\left( x \right) d x\]
\[ \Rightarrow I = \int_0^\frac{\pi}{2} f\left( x \right) d x + \int_\frac{\pi}{2}^3 f\left( x \right) d x + \int_3^9 f\left( x \right) d x ....................\left[\text{Additive property} \right]\]
\[ \Rightarrow I = \int_0^\frac{\pi}{2} \sin x d x + \int_\frac{\pi}{2}^3 1 d x + \int_3^9 e^{x - 3} d x\]
\[ \Rightarrow I = \left[ - \cos x \right]_0^\frac{\pi}{2} + \left[ x \right]_\frac{\pi}{2}^3 + \left[ e^{x - 3} \right]_3^9 \]
\[ \Rightarrow I = 0 + 1 + 3 - \frac{\pi}{2} + e^6 - e^0 \]
\[ \Rightarrow I = 3 - \frac{\pi}{2} + e^6\]