Question

Find the area bounded by the $X$ -axis the parabola $y=-x^2+4$ .

Answer 1

Given $y=-x^2+4$

Now

$A={\int }_{x_1}^{x_2}({-x}^2+4)dx={\int }_{-2}^2({-x}^2+4)$

$=\frac{x^3}{3}+4x{\int }_{-2}^2=\frac{{-2}^3}{3}+4\left(2\right)-(\frac{{-(-2)}^3}{3}+4(-2))$

$=\frac{2^3}{3}+8-(\frac{2^3}{3}-8)$

$A=|\frac{-2^3}{3}+8\frac{-2^3}{3}+8|=|16-\frac{16}{3}|=\frac{32}{3}$