Answer
$\dfrac{\sqrt 2}{3}$
Work Step by Step
$I=\int_{0}^{\pi/2} (\cos t)^2 (\sin t) \sqrt {{(-\sin t)^{2}}+(\cos t)^2 +1^2 }dt$
$=\int_{0}^{\pi/2} (\cos t)^2 (\sin t) (\sqrt 2)dt$
$=\sqrt 2 \int_{0}^{\pi/2} \cos^2 t(\sin t) dt$
Plug $a=\cos t $ and $da=-\sin t dt$
$=-\sqrt 2 \int_{1}^{0} a^2 da$
$=-\sqrt 2[\dfrac{a^3}{3}]_{1}^{0}$
$=\dfrac{\sqrt 2}{3}$
