Calculadora de integrales definidas e impropias
Calcular integrales definidas e impropias paso a paso
La calculadora intentará evaluar la integral definida (es decir, con límites de integración), incluyendo las impropias, mostrando los pasos.
Solution
Your input: calculate $$$\int_{\sqrt{6}}^{4}\left( 6 - x^{2} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\left(6 - x^{2}\right)d x}=\frac{x \left(18 - x^{2}\right)}{3}$$$ (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.
$$$\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=4\right)}=\frac{8}{3}$$$
$$$\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=\sqrt{6}\right)}=4 \sqrt{6}$$$
$$$\int_{\sqrt{6}}^{4}\left( 6 - x^{2} \right)dx=\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=4\right)}-\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=\sqrt{6}\right)}=\frac{8}{3} - 4 \sqrt{6}$$$
Answer: $$$\int_{\sqrt{6}}^{4}\left( 6 - x^{2} \right)dx=\frac{8}{3} - 4 \sqrt{6}\approx -7.13129230446605$$$