Calculadora de Integrais Definidas e Impróprias

A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.

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Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

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Solution

Your input: calculate $$$\int_{0}^{8}\left( - 2 x^{2} + 16 x \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(- 2 x^{2} + 16 x\right)d x}=\frac{2 x^{2} \left(12 - x\right)}{3}$$$ (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.

$$$\left(\frac{2 x^{2} \left(12 - x\right)}{3}\right)|_{\left(x=8\right)}=\frac{512}{3}$$$

$$$\left(\frac{2 x^{2} \left(12 - x\right)}{3}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{8}\left( - 2 x^{2} + 16 x \right)dx=\left(\frac{2 x^{2} \left(12 - x\right)}{3}\right)|_{\left(x=8\right)}-\left(\frac{2 x^{2} \left(12 - x\right)}{3}\right)|_{\left(x=0\right)}=\frac{512}{3}$$$

Answer: $$$\int_{0}^{8}\left( - 2 x^{2} + 16 x \right)dx=\frac{512}{3}\approx 170.666666666667$$$

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Calculadora de Integrais Definidas e Impróprias

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Solution