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 $$$\int_{0}^{2}\left( \pi \left(- x^{2} + 2 x\right) \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\pi \left(- x^{2} + 2 x\right) d x}=\frac{\pi x^{2} \left(3 - x\right)}{3}$$$ (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.

$$$\left(\frac{\pi x^{2} \left(3 - x\right)}{3}\right)|_{\left(x=2\right)}=\frac{4 \pi}{3}$$$

$$$\left(\frac{\pi x^{2} \left(3 - x\right)}{3}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{2}\left( \pi \left(- x^{2} + 2 x\right) \right)dx=\left(\frac{\pi x^{2} \left(3 - x\right)}{3}\right)|_{\left(x=2\right)}-\left(\frac{\pi x^{2} \left(3 - x\right)}{3}\right)|_{\left(x=0\right)}=\frac{4 \pi}{3}$$$

Answer: $$$\int_{0}^{2}\left( \pi \left(- x^{2} + 2 x\right) \right)dx=\frac{4 \pi}{3}\approx 4.18879020478639$$$

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Solution