Answer
(a) $\frac{25}{2}$
(b) 30
(c) $\frac{21}{2}$
Work Step by Step
Evaluate definite integral
(a) $\int_{0}^{5} x d x=\left.\frac{x^{2}}{2}\right|_{0} ^{5}=-\frac{0}{2}+\frac{25}{2}=\frac{25}{2}$
Using geometry
\[
\int_{0}^{5} x d x=\frac{5 \cdot 5}{2}=\frac{25}{2}
\]
Evaluate definite integral
(b) $\int_{3}^{9} 5 d x=\left.(5 x)\right|_{3} ^{9}=-15+45=30$
Using geometry
$\int_{3}^{9} 5 d x=5 \cdot 6=30$
Evaluate definite integral
(c) $\int_{-1}^{2}(3+x) d x=\left.\left(\frac{x^{2}}{2}+3 x\right)\right|_{-1} ^{2}=\left(6+\frac{4}{2}\right)-$
$\left(\frac{1}{2}-3\right)=\frac{21}{2}$
Using geometry
$\int_{-1}^{2}(3+x) d x=\frac{(5+2) \cdot 3}{2}=\frac{21}{2}$
