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## 'Area L' printed from http://nrich.maths.org/

### Why do this problem?

This
problem is not nearly as complicated as it first appears.
Students need to interpret the notation and use their understanding
of definite integrals as areas under curves to prove a formula
graphically. Once they make sense of the formula in terms of a
diagram, they can use it to evaluate integrals which might prove
challenging to calculate by other means.

### Possible approach

Students could sketch a suitable continuous increasing function on
paper or whiteboards, and mark on suitable points $a, b, f(a)$ and
$f(b)$. Give them time in groups to identify the areas on their
graph represented by $\int_{f(a)}^{f(b)} f^{-1}(t)dt$,
$\int_a^bf(x)dx$, $bf(b)$ and $af(a)$.

Once they are convinced that the proof of the formula follows from
a diagram, give them the chance to verify the formula for cases
they can integrate directly, like $\int_a^b \sqrt{t}dt$.

The problem suggests using the formula to evaluate $\int_0^1
\sin^{-1}tdt$ but other inverse functions of continuous increasing
functions can be used, and it offers good integration practice to
allow students to suggest functions of their own to integrate which
they can evaluate by other means as a check. It is worth taking the
time to discuss the requirement for the function to be increasing
between $a$ and $b$, and setting the challenge to find a similar
formula for a function which is decreasing.

### Key questions

On the graph $y=f(x)$, what is represented by
$\int_{f(a)}^{f(b)} f^{-1}(t)dt$?

Why does the formula specify that $f(x)$ is increasing?

What could be done if $f(x)$ were decreasing?

### Possible extension

Integral
Sandwich is another problem where sketching a graph makes the
meaning of the integrals much clearer.

### Possible support

Sketching some particular inverse functions like $y=\sqrt x$ and
marking on numerical values for a and b, then working out the
relevant areas might provide a way into this problem.