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The integral describes the area of a semicircle of radius $k$ . The given expression is then equal to

$\displaystyle \dfrac{1}{\pi} \sum_{k=1}^{20} \dfrac{\pi k^{2}}{2} = \dfrac{1}{2} \sum_{k=1}^{20} k^{2} = \dfrac{1}{2} \times \dfrac{20 \times (20 + 1) \times (2 \times 20 + 1)}{6} = \boxed{1435}$ .