Đáp án:
\(I = \dfrac{4\pi}{3} - \dfrac{3\sqrt3}{2}\)
Giải thích các bước giải:
\(\begin{array}{l}
\quad I = \displaystyle\int\limits_0^3\left(\sqrt{4x-x^2} - \sqrt x\right)dx\\
\Leftrightarrow I = \displaystyle\int\limits_0^3\sqrt{4x- x^2}dx - \displaystyle\int\limits_0^3\sqrt x\ dx\\
\Leftrightarrow I = \displaystyle\int\limits_0^3\sqrt{4 - (x-2)^2}\ dx - \displaystyle\int\limits_0^3\sqrt x\ dx\\
\Leftrightarrow I = \left[\dfrac{(x-2)\sqrt{4 - (x-2)^2}}{2} + 2\arcsin\dfrac{x-2}{2}\right]\Bigg|_0^3\ -\ \dfrac23\sqrt{x^3}\ \Bigg|_0^3\\
\Leftrightarrow I = \dfrac{\sqrt3}{2} + \dfrac{4\pi}{3} - 2\sqrt3\\
\Leftrightarrow I = \dfrac{4\pi}{3} - \dfrac{3\sqrt3}{2}
\end{array}\)
