정적분 및 가적분 계산기

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Solution

Your input: calculate $$$\int_{3}^{11}\left( \frac{y^{\frac{3}{2}}}{3} - \sqrt{y} \right)dy$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}\right)d y}=\frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15}$$$ (for steps, see indefinite integral calculator)

According to the Fundamental Theorem of Calculus, $$$\int_a^b F(x) dx=f(b)-f(a)$$$, so just evaluate the integral at the endpoints, and that's the answer.

$$$\left(\frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15}\right)|_{\left(y=11\right)}=\frac{44 \sqrt{11}}{5}$$$

$$$\left(\frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15}\right)|_{\left(y=3\right)}=- \frac{4 \sqrt{3}}{5}$$$

$$$\int_{3}^{11}\left( \frac{y^{\frac{3}{2}}}{3} - \sqrt{y} \right)dy=\left(\frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15}\right)|_{\left(y=11\right)}-\left(\frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15}\right)|_{\left(y=3\right)}=\frac{4 \sqrt{3}}{5} + \frac{44 \sqrt{11}}{5}$$$

Answer: $$$\int_{3}^{11}\left( \frac{y^{\frac{3}{2}}}{3} - \sqrt{y} \right)dy=\frac{4 \sqrt{3}}{5} + \frac{44 \sqrt{11}}{5}\approx 30.5719388011826$$$

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Solution