Question

Calcule o valor da área limitada pela função y = 2x – x^2 e pelo eixo dos X no intervalo 0 x 2.

Answer 1

Boa noite Darlene!

Solução!

$\boxed{A=\displaystyle \int_{a}^{b}f(x)dx=f(a)-f(b)\bigg|_{a}^{b}}\\\\\\ A=\displaystyle \int_{0}^{2}(2x- x^{2}) dx=2\bigg( \frac{ x^{2} }{2}- \frac{ x^{3} }{3}\bigg)\bigg|_{0}^{2} \\\\\\\\ A=\bigg(x^{2} - \frac{ x^{3} }{3}\bigg)\bigg|_{0}^{2}\\\\\\ A=\bigg(x^{2} - \dfrac{ x^{3} }{3}\bigg)-\bigg(x^{2} - \dfrac{ x^{3} }{3}\bigg)\\\\\\\ A=\bigg((2)^{2} - \dfrac{ (2)^{3} }{3}\bigg)-\bigg((0)^{2} - \dfrac{ (0)^{3} }{3}\bigg)\\\\\\\ A=\bigg(4 - \dfrac{ 8 }{3}\bigg)\\\\\\\ A=\bigg(\dfrac{12-8 }{3}\bigg)$

$A=\bigg(\dfrac{4 }{3}\bigg)$

$\boxed{Resposta:~~\displaystyle \int_{0}^{2}(2x- x^{2}) dx= \frac{2}{3}u~a}$

Boa noite!
Bons estudos!