Area between curves- 1 Dec'15

Question

Find approx area of the region between curves !!y = x^2!!, !!y = |2-x^2|!! and !!y = 2!! which lies right of the line !!x = 1!!.

!!a)" "1/2!!

!!b)" "1!!

!!c)" "3/2!!

!!d)" "!!None

Solution

The given curves !!y = x^2!!, !!y = |2-x^2|!! and !!y = 2!! can be traced on x-y cordinate as:

We need to find the area between the curves right of line !!x = 1!!.

Area = !!int_sqrt2\^2 2dx + int_1\^sqrt2 [x^2 - (2-x^2)]dx - int_sqrt2\^2 (x^2 - 2)dx!!

!!=> Area = 2(2-sqrt2) + [(x^3)/3 - 2x + (x^3)/3]_1\^sqrt2 + [2x - (x^3)/3]_sqrt2\^2!!

!!= 4 - 2sqrt2 + (4sqrt2)/3 - 2/3 - 2sqrt2 + 2 + 4/3 - (4sqrt2)/3!!

!!= 20/3 - 4sqrt2!!

!!= (20-17)/3!!

!!= 1!!