定積分・広義積分計算機
定積分と広義積分を手順を追って計算する
この計算機は、定積分（すなわち上下限のある積分）を、広義積分も含めて、解法手順を示しながら計算を試みます。
Solution
Your input: calculate $$$\int_{0}^{3}\left( \sqrt{9 - x^{2}} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\sqrt{9 - x^{2}} d x}=\frac{x \sqrt{9 - x^{2}}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{2}$$$ (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.
$$$\left(\frac{x \sqrt{9 - x^{2}}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{2}\right)|_{\left(x=3\right)}=\frac{9 \pi}{4}$$$
$$$\left(\frac{x \sqrt{9 - x^{2}}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{2}\right)|_{\left(x=0\right)}=0$$$
$$$\int_{0}^{3}\left( \sqrt{9 - x^{2}} \right)dx=\left(\frac{x \sqrt{9 - x^{2}}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{2}\right)|_{\left(x=3\right)}-\left(\frac{x \sqrt{9 - x^{2}}}{2} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{2}\right)|_{\left(x=0\right)}=\frac{9 \pi}{4}$$$
Answer: $$$\int_{0}^{3}\left( \sqrt{9 - x^{2}} \right)dx=\frac{9 \pi}{4}\approx 7.06858347057703$$$