Question

Find the area of the figure bounded by the following curves
Find all positive a which satisfy the condition ${\int }_0^a(3x^2+4x-5)dx=a^3-2.$

Answer 1

${\int }_0^a(3x^2+4x-5)dx=a^3-2$

$⟹3{\left[\frac{x^3}{3}\right]}_0^a+4{\left[\frac{x^2}{2}\right]}_0^a-5[x{]}_0^a=a^3-2$

$⟹a^3+2a^2-5a=a^3-2$

$⟹2a^2-5a+2=0$

$⟹(2a-1)(a-2)=0$

$⟹a=\frac{1}{2},2$