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This is a problem from ISI MStat 2019 PSA Problem 16 based on calculating area bounded by the curve.

## Area bounded by the curve – ISI MStat 2018 PSA Problem 16

The functions $f, g:[0,1] \rightarrow[0,1]$ are given by $f(x)=\frac{1}{2} x(x+1)$ and $g(x)=\frac{1}{2} x^{2}(x+1) .$ What is the area enclosed between the graphs of $f^{-1}$ and $g^{-1} ?$

• (A) 1/8
• (B) 1/4
• (C)5/12
• (D) 7/24

### Key Concepts

Integration

Graph of a function

## Check the Answer

But try the problem first…

Source

ISI MStat 2019 PSA Problem 16

Introduction to Real Analysis by Bertle Sherbert

## Try with Hints

First hint

Inverse of a function is basically reflection about y=x line .

So , we can get $f^{-1}$ and $g^{-1}$ from $f(x)$ and $g(x)$ respectively by replacing x by y and y by x .

Let’s draw the curves .

Second Hint

This is graph of inverses of f and g when they are defined in $R \to R$ . But in our problem we should consider one positive x axis and y axis .

Final Step

Therefore area of the curve bounded by the graphs of $f^{-1}$ and $g^{-1}$ is $\int^1_{0}\frac{y^2+y-y^3-y^2}{2} \,dx$ = 1/8 .