Answer
a) 2 and b) $\dfrac{2a}{3}$
Work Step by Step
a) Average value can be calculated as: $\dfrac{1}{3} \int_{0}^3 \sqrt {3x} dx$
$=\dfrac{\sqrt 3}{3} [\dfrac{x^{3/2}}{3/2}]_{0}^3$
or, $\dfrac{2\sqrt 3}{9} [3^{3/2}-0]=2$
b) Average value can be calculated as: $\dfrac{1}{a} \int_{0}^a \sqrt {ax} dx$
$=\dfrac{\sqrt a}{a} [\dfrac{x^{3/2}}{3/2}]_{0}^a$
or, $\dfrac{2\sqrt a}{3a} [3^{a/2}-0]=\dfrac{2a}{3}$
Hence, a) 2 and b) $\dfrac{2a}{3}$
