Question

Find the area of the region bounded by the curves

y=x^2-4x+4 and y=10-x^2

from x=10 and x=4

Answer 1

Given curves are $y=x^2-4x+4 ; y=10-x^2$

Graph of the above curves are:

So, area bounded by these curves from $x=4$ to $x=10$ will be given as:

$∫_4^{10} (x^2-4x+4) dx +|∫_4^{10} (10-x^2)dx| \\ =∫_4^{10} (x-2)^2 dx + |[10x-\frac{x^3}{3}]_4^{10}| \\ = [\frac{(x-2)^3}{3}]_4^{10} + |100-\frac{1000}{3}-40+\frac{64}{3}| \\ =\frac{512}{3}-\frac{64}{3}+|60-\frac{936}{3}|\\ =\frac{448}{3}+\frac{756}{3}\\ =\frac{1204}{3}\\ =401.33 \ sq. units$