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Solve this integral equation.

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Generalise this inequality involving integrals.

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Integral Inequality

An inequality involving integrals of squares of functions.

Area L

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

 

Well done to Andrei from Tudor Vianu for his solution for the first question.

1) Draw a diagram to prove: $\int_{f(a)}^{f(b)} f^{-1}(t)dt + \int_a^bf(x)dx = bf(b) - af(a)$

To prove the formula, I represented a sketch of the graph of a function f(x):

 

Now, I identify the parts of the formula on this graph:

1. $\int_a^bf(x)dx$ is the area between the x-axis and the graph, on the domain of x between a and b. This is the green area below.

2. $\int_{f(a)}^{f(b)} f^{-1}(t) dt$ is the area between the y axis and the part of the graph situated between f(a) and f(b). This is yellow below.
This becomes more evident by by considering the graph of $f^{-1}(t)$, which is the above, but reflected in the line $y=x$

3. $bf(b)$ and $af(a)$ are the areas of the rectangles in my diagram, of sides $b, f(b)$, and $a, f(a)$ respectively.

The difference of the two areas is exactly the sum of the yellow and green areas.

Derek Wan continues to finish the problem:

2) Find the value of $\int _1^4 \sqrt t dt$, firstly by evaluating the integral directly, and secondly by using the formula above with $f(x)=x^2$.

 

 

3) Use the formula to evaluate $\int_0^1\sin^{-1}t dt$


The last few ideas remain unanswered, can you add to them?

What other functions can you integrate more easily using this formula than by other means? Why must $f(x)$ be increasing in the interval $a\leq x\leq b$? How could you evaluate a similar integral if $f(x)$ is decreasing?