Question

(i) $\underset{1}{\overset{4}{\int }}f\left(x\right)dx,\mathrm{where}f\left(x\right)=\left\{\begin{array}{l}4x+3,\mathrm{if}1\le x\le 2\\ 3x+5,\mathrm{if}2\le x\le 4\end{array}\right.$

(ii) $\underset{0}{\overset{9}{\int }}f\left(x\right)dx,\mathrm{where}f\left(x\right)\left\{\begin{array}{ccc}\sin x& ,& 0\le x\le \mathrm{\pi }/2\\ 1& ,& \mathrm{\pi }/2\le \mathrm{x}\le 3\\ e^{x-3}& ,& 3\le x\le 9\end{array}\right.$

(iii) $\underset{1}{\overset{4}{\int }}f\left(x\right)dx,\mathrm{where}f\left(x\right)=\left\{\begin{array}{ccc}7x+3& ,& \mathrm{if}1\le x\le 3\\ 8x& ,& \mathrm{if}3\le x\le 4\end{array}\right.$

Answer 1

(i) We have,
$\underset{1}{\overset{4}{\int }}f\left(x\right)dx,\mathrm{where}f\left(x\right)=\left\{\begin{array}{l}4x+3,\mathrm{if}1\le x\le 2\\ 3x+5,\mathrm{if}2\le x\le 4\end{array}\right.$

$$\begin{gathered} I={\int }_1^{4}f\left(x\right)\mathit{d}x \\ \Rightarrow I={\int }_1^{2}f\left(x\right)\mathit{}\mathit{d}x+{\int }_2^{4}f\left(x\right)\mathit{d}x\left[\mathrm{Additive}\mathrm{property}\right] \\ \Rightarrow I={\int }_1^2\left(4x+3\right)dx+{\int }_2^4\left(3x+5\right)dx \\ \Rightarrow I={\left[2x^2+3x\right]}_1^2+{\left[\frac{3x^2}{2}+5x\right]}_2^4 \\ \Rightarrow I=8+6-2-3+24+20-6-10 \\ \Rightarrow I=37 \end{gathered}$$

(ii) We have,
$\underset{0}{\overset{9}{\int }}f\left(x\right)dx,\mathrm{where}f\left(x\right)\left\{\begin{array}{ccc}\sin x& ,& 0\le x\le \frac{\mathrm{\pi }}{2}\\ 1& ,& \frac{\mathrm{\pi }}{2}\le \mathrm{x}\le 3\\ e^{x-3}& ,& 3\le x\le 9\end{array}\right.$

$$\begin{gathered} I={\int }_0^{9}f\left(x\right)\mathit{d}x \\ \Rightarrow I={\int }_0^{\frac{\mathrm{\pi }}{2}}f\left(x\right)\mathit{d}x+{\int }_{\frac{\mathrm{\pi }}{2}}^{3}f\left(x\right)\mathit{d}x+{\int }_3^{9}f\left(x\right)\mathit{d}x\mathit{}\left[\mathrm{Additive}\mathrm{property}\right] \\ \Rightarrow I={\int }_0^{\frac{\mathrm{\pi }}{2}}\sin x\mathit{d}x+{\int }_{\frac{\mathrm{\pi }}{2}}^{3}1\mathit{d}x+{\int }_3^9{e}^{x-3}\mathit{d}x \\ \Rightarrow I={\left[-\cos x\right]}_0^{\frac{\mathrm{\pi }}{2}}+{\left[x\right]}_{\frac{\mathrm{\pi }}{2}}^3+{\left[e^{x-3}\right]}_3^9 \\ \Rightarrow I=0+1+3-\frac{\mathrm{\pi }}{2}+e^6-e^0 \\ \Rightarrow I=3-\frac{\mathrm{\pi }}{2}+e^6 \end{gathered}$$

(iii) We have,
$\underset{1}{\overset{4}{\int }}f\left(x\right)dx,\mathrm{where}f\left(x\right)=\left\{\begin{array}{ccc}7x+3& ,& \mathrm{if}1\le x\le 3\\ 8x& ,& \mathrm{if}3\le x\le 4\end{array}\right.$

$$\begin{gathered} I={\int }_1^{4}f\left(x\right)\mathit{}\mathit{d}x \\ \Rightarrow I={\int }_1^{3}f\left(x\right)\mathit{d}x+{\int }_3^{4}f\left(x\right)\mathit{d}x\left[\mathrm{Additive}\mathrm{property}\right] \\ \Rightarrow I={\int }_1^3\left(7\mathrm{x}+3\right)\mathit{}\mathit{d}x+{\int }_3^{4}8x\mathit{d}x \\ \Rightarrow I={\left[\frac{7{\mathrm{x}}^2}{2}+3x\right]}_1^3+{\left[4x^2\right]}_3^4 \\ \Rightarrow I=\frac{63}{2}+9-\frac{7}{2}-3+64-36 \\ \Rightarrow I=\frac{56}{2}+34 \\ \Rightarrow I=62 \end{gathered}$$