Calcolatore di integrali definiti e impropri

Il calcolatore cercherà di valutare l'integrale definito (cioè con estremi), inclusi quelli impropri, mostrando i passaggi.

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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: $$$\int_{0}^{3}\left( \sqrt{9 - x^{2}} \right)dx=\frac{9 \pi}{4}\approx 7.06858347057703$$$

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Solution