Answer
$\int_C \textbf F \cdot \textbf{dr} = -\frac{16}{3}+\frac{\pi^6}{24}$
Work Step by Step
$\textbf r(t) = 2\sin t \textbf i + 2\cos t \textbf j +\frac{1}{2}t^2 \textbf k$
it follows that, $x(t) = 2\sin t , y(t) = 2\cos t, z(t) = \frac{1}{2}t^2 $
$\textbf F(x,y,z) = x^2 \textbf i +y^2\textbf j + z^2\textbf k\\
\therefore \textbf F(x(t),y(t),z(t)) = 4\sin^2t \textbf i + 4\cos^2t \textbf j +\frac{t^2}{4}\textbf k\\
\textbf r' = 2\cos t \textbf i - 2\sin t \textbf j +t \textbf k\\
\int_C \textbf F \cdot d\textbf r = \int_{t=0}^{\pi}\textbf F \cdot \textbf r' dt\\
= \int_0^{\pi}(4\sin^2t \textbf i + 4\cos^2t \textbf j +\frac{t^2}{4}\textbf k)\cdot (2\cos t \textbf i - 2\sin t \textbf j +t \textbf k) dt\\
= \int_0^{\pi} (8\sin^2t.\cos t -8\cos^2t.\sin t + \frac{t^5}{4}) dt\\
=[\frac{8\sin^3t}{3}+\frac{8\cos^3t}{3}+\frac{t^6}{24}]_0^{\pi} = -\frac{16}{3}+\frac{\pi^6}{24}$
