Question

Find the area enclosed between the curve $y=5x-x^{2}andy=x$ .

• A. $16$
• B. $32$
• C. $\frac{32}{3}$
• D. $\frac{32}{9}$

Answer 1

The correct option is C

$\frac{32}{3}$

Let’s have a look at the graph of both the functions.

We can see that both the function intersect each other at points (0,0) and (4,4). And the area enclosed will be the area between these points.
Let the area enclosed be A.
$A={\int }_0^4(5x-x^2)-xdx$
Since, throughout the interval $[0,4],5x-x^2\ge x$
So we don’t have to split the integral.
$A={\int }_0^{4}4x-x^{2}dxA=(2x^2-\frac{x^3}{3}{)}_0^{4}A=32-\frac{64}{3}A=\frac{32}{3}$