Question

Evaluate: ${\int }_1^3(2x^2+5x-1)dx$ as limit of sums.

Answer 1

We have ${\int }_a^{b}f\left(x\right)dx=f\left(b\right)-f\left(a\right)$

${\int }_1^3(2x^2+5x-1)dx$

$={[\frac{2x^3}{3}-\frac{5x^2}{2}-x]}_1^3$

$=(\frac{2\times 3^3}{3}-\frac{5\times 3^2}{2}-3)-(\frac{2\times 1^3}{3}-\frac{5\times 1^2}{2}-1)$

$=(18-\frac{45}{2}-3)-(\frac{2}{3}-\frac{5}{2}-1)$

$=\frac{-15}{2}+\frac{17}{6}$

$=\frac{-45+17}{6}$

$=\frac{-28}{6}$

$=\frac{-14}{3}$