Question

Find the area of the figure bounded by the following curves
Find all values of a for which the inequality ${\int }_0^{a}xdx⩽a+4$ is satisfied.

Answer 1

${\int }_0^{a}xdx\le a+4$

$\Rightarrow \frac{x^2}{2}{|}_0^a\le a+4$

$\Rightarrow \frac{a^2}{2}\le a+4$

$\Rightarrow a^2\le 2a+8$

$\Rightarrow a^2-2a-8\le 0$

$\Rightarrow a^2-4a+2a-9\le 0$

$\Rightarrow a(a-4)+2(a-4)\le 0$

$\Rightarrow (a+2)(a-4)\le 0$

Compute the signs : -

(i) $a\le -2$

$a<-2:(a+2)(a-4)>0$

$a=-2:(a+2)(a-4)=0$

(ii) $-2<a<4$

$(a+2)(a-4)<-$

(iii) $a\ge 4$

$a=5:(a+2)(a-4)=0$

$a>4:(a+2)(a-4)>0$

Choosing the ranges that satisfy the required condition :-

Solution : - $-2\le a\le 4$.