Answer
(b) 31
(a) $\frac{263}{4}$
Work Step by Step
We know:
$f_{\text {ave}}=\frac{1}{-a+b} \int_{a}^{b} f(x) d x$
(a) $v_{\text {ave }}=\frac{1}{3} \int_{1}^{4}\left(2+3 t^{3}\right) d t= \frac{789}{4}\cdot \frac{1}{3} =\frac{263}{4}$
$2+3 t^{3}=f(x), b=4$ and $a=1$
$2 t+\frac{3}{4} t^{4}=F(x)$
Using the Fundamental Theorem of Calculus
(b) $v_{a v e}=\frac{1}{b-a} \int_{a}^{b} v(x) d x=\frac{1}{b-a}(s(b)-s(a))$
$31=(-7+100)\frac{1}{3}=v_{a v e}$
Evaluate at $b=4$ and $a=1$
