Calculadora de Integrais Definidas e Impróprias
Calcule integrais definidas e impróprias passo a passo
A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.
Solution
Your input: calculate $$$\int_{20}^{90}\left( \frac{312 \pi \left(4 - \frac{4 x}{9}\right)^{2}}{5} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\frac{312 \pi \left(4 - \frac{4 x}{9}\right)^{2}}{5} d x}=\frac{1664 \pi x \left(x^{2} - 27 x + 243\right)}{405}$$$ (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.
$$$\left(\frac{1664 \pi x \left(x^{2} - 27 x + 243\right)}{405}\right)|_{\left(x=90\right)}=2186496 \pi$$$
$$$\left(\frac{1664 \pi x \left(x^{2} - 27 x + 243\right)}{405}\right)|_{\left(x=20\right)}=\frac{685568 \pi}{81}$$$
$$$\int_{20}^{90}\left( \frac{312 \pi \left(4 - \frac{4 x}{9}\right)^{2}}{5} \right)dx=\left(\frac{1664 \pi x \left(x^{2} - 27 x + 243\right)}{405}\right)|_{\left(x=90\right)}-\left(\frac{1664 \pi x \left(x^{2} - 27 x + 243\right)}{405}\right)|_{\left(x=20\right)}=\frac{176420608 \pi}{81}$$$
Answer: $$$\int_{20}^{90}\left( \frac{312 \pi \left(4 - \frac{4 x}{9}\right)^{2}}{5} \right)dx=\frac{176420608 \pi}{81}\approx 6842489.951045$$$