Question

${\int }_1^{4}f\left(x\right)dx$, find this where $f\left(x\right)=\{\begin{array}{cc}7x+3& 1\le x\le 3\\ 8x& 3\le x\le 4\end{array}$

Answer 1

$I={\int }_1^{4}f\left(x\right)dx$

$I={\int }_1^3(7x+3)dx+{\int }_3^4\left(8x\right)dx$

$I={[\frac{7x^2}{2}+3x]}_1^3+{\left[\frac{8x^2}{2}\right]}_3^4$

$I=\frac{7}{2}[9-1]+3[3-1]+4[16-9]$

$I=\frac{7}{2}\times 8+3\left(2\right)+4\left(7\right)$

$I=28+6+28$

$I=62$.