The area bounded by the two branches of curve `(y-x)^2=x^3` and the straight line `x=1` is (a) `1/5s qdotu n i t s` (b) `3/5s qdotu n i t s` `4/5s qdotu n i t s` (d) `8/4s qdotu n i t s`

Options

A. `1//5` sq. units
B. `3//5` sq. units
C. `4//5` sq. units
D. `8//4` sq. units

Answer

C

Explanation

`(y-x)^(2)=x^(3)," where "xge 0 rArry-x = pm x^(3//2)`
`"or "y=x+x^(3//2)" (1)"`
`y=x-x^(3//2)" (2)"`
Function (1) is an increasing function.
Function (2) meets x-axis, when `x-x^(3//2)=0 or x=0, 1`.
`"Also, for "0ltxlt1,x-x^(3//2)gt0 and" for "xgt1,x-x^(3//2)lt0.`
`"When "xrarroo,x-x^(3//2)rarr-oo`.
From these information, we can plot the graph as shown.

`"Required area "=overset(1)underset(0)int[(x+x^(3//2))-(x-x^(3//2))]dx`
`=2overset(1)underset(0)intx^(3//2)dx`
`=2[(x^(5//2))/(5//2)]_(0)^(1)=(4)/(5)`sq. units.