Question

a area da regiao limitada pelo eixo dos x e pelo gráfico de y=4-x^2 é : opacao a ) 33/2. b ) 32/3. c)32/5. d)33/4. e)32/7

Answer 1

$\mathrm{y=4-x^2}\\\\ \textbf{Abscissas de intersec\c{c}\~ao:}\\\\ \mathrm{0=4-x^2\ \to\ x^2=4\ \to\ x=\pm\sqrt{4}\ \to\ x=\pm2}\\\\ \textbf{\'Area entre a curva e o eixo das abscissas:}\\\\ \mathrm{S=\int\limits_{-2}^24-x^2\ dx=\bigg(\int4\ dx-\int x^2\ dx\bigg)\bigg|_{-2}^2=}\\\\\\ \mathrm{=\bigg(4x-\dfrac{x^3}{3}\bigg)\bigg|_{-2}^2=\bigg(\dfrac{12x-x^3}{3}\bigg)\bigg|_{-2}^2=}\\\\\\ \mathrm{=\bigg(\dfrac{12.2-2^3}{3}\bigg)-\bigg(\dfrac{12.(-2)-(-2)^3}{3}\bigg)=}$

$\mathrm{=\bigg(\dfrac{24-8}{3}\bigg)-\bigg(\dfrac{-24-(-8)}{3}\bigg)=}\\\\\\ \mathrm{=\dfrac{16}{3}-\bigg(\dfrac{-16}{3}\bigg)=\dfrac{16}{3}+\dfrac{16}{3}=\dfrac{32}{3}}\\\\\\ \mathbf{b)}\ \boxed{\boxed{\mathbf{S=\dfrac{32}{3}\ u.a.}}}$