Question

Calcule a área sob o gráfico da função f(x)=-x^2+3x no intervalo [0,3] .

a) -27/2
b) 27/2
c) 9/2
d) -9/2
e) 45/2

Answer 1

$\int\limits^3_0 {-x^2 + 3x} \, dx \\\int\limits^3_0 {-x^2} \, dx + \int\limits^3_0 {3x} \, dx\\(-1) * \int\limits^3_0 {x^2} \, dx + 3 * \int\limits^3_0 {x} \, dx\\(-1) * \frac{x^3}{3} _{x = 3, x=0} + 3 * \frac{x^2}{2}_{x = 3, x=0} \\(-1) * [\frac{3^3}{3} - \frac{0^3}{3} ] + 3 * [\frac{3^2}{2} - \frac{0^2}{2} ]\\(-1) * [\frac{3^3}{3}] + 3 * [\frac{3^2}{2}]\\(-1) * [3^2] + 3 * [\frac{9}{2}]\\-9 + \frac{27}{2} = \frac{-18 + 27}{2} = \frac{9}{2}$