P164. Show that the area of the bounded region enclosed between the curves (y^3=x^2) and (y=2-x^2), is (2\frac{2}{15}).

Solution:

Note that (y=x^{\frac{2}{3}}) is an even function (green line).

P165. Find the area of the region in the xy plane, bounded by the graphs of (y=x^2), x+y = 2 and (y=-\sqrt {x})

Solution:

The parabola and straight line intersects at (1,1) (we find that by solving the (y=x^2) and x+y=2)
Thus the area is found by adding area under parabola (from 0 to 1) and area under straight line (from 1 to 2).
(\int^1_0 x^2,dx=\left[\frac{x^3}{3}\right]^1_0=\frac{1}{3}) (area under parabola)
area under straight line above ‘x’ axis is the triangle with height 1 unit and base 1 unit (from x=1 to x=2, area under x+y=2)
that area = (\frac{1}{2}\times 1\times 1=\frac{1}{2})
Thus total area above x axis (of the required region) is (\frac{1}{2}+\frac{1}{3}=\frac{5}{6})
Now we come to the region below ‘x’ axis.
x+y = 2 and (y=-\sqrt{x}) intersect at (4, -2) (found by solving the two equations). We calculate the area under the curve (y=-\sqrt{x}) from x=0 to x=4 and subtract from it the area of the triangle with base from x=2 to x=4 and height =2 (hence the area of the triangle to be subtracted is 2 sq unit).
Area under the square root curve is
(|\int^4_0 -\sqrt{x},dx|=\int^4_0 \sqrt{x},dx= \left[\frac {x^{\frac{1}{2}+1}} {\frac{1}{2}+1}\right]^4_0).
=(\frac {2}{3}\times 8=\frac {16}{3})
Delete 2 square unit from this and add the area computed before (above ‘x’ axis).
Area = (\frac {16}{3} - 2 + \frac{5}{6} = \frac{25}{6}) (ANS)