# Definite Integrals with inverse of function

In summary, the problem asks to find the integral of f^-1(x) from 5 to 9, given that f(x) is continuous and decreasing on the interval 5 <= x < 13 with f(5) = 9, f(13) = 5, and the integral of f(x) from 5 to 13 is 70.64758. To solve this, we can use the formula \int_{\alpha}^{\beta} f^-1(y) dy = b \beta - a \alpha - \int_a^b f(x) dx, where f is a positive monotone function on [a,b] and has an inverse f^-1. By setting \alpha =

## Homework Statement

Suppose f(x) is continuous and decreasing on the closed interval 5 <= x < 13, that f(5) = 9, f(13) = 5 and that the

integral of f(x) from 5 to 13 is 70.64758.

Then the integral of f^-1(x) from 5 to 9 is equal to what?

(Note: f^-1(x) is the inverse of f(x))

## The Attempt at a Solution

I really don't know how to solve this problem. I know that f^-1(f(x)) = x. Any ideas would be great. Thanks.

You want to find $$\int_{u=5}^9 f^{-1}(u)\,du$$

Try the substitution u=f(x)

f is a positive monotone function on [a,b] where 0 < a < b and f has an inverse $$f^-1$$. Set $$\alpha = f(a), \beta = f(b)$$ and then use this formula:$$\int_{\alpha}^{\beta} f^-1(y) dy = b \beta - a \alpha - \int_a^b f(x) dx$$.

JG89 said:
f is a positive monotone function on [a,b] where 0 < a < b and f has an inverse $$f^-1$$. Set $$\alpha = f(a), \beta = f(b)$$ and then use this formula:$$\int_{\alpha}^{\beta} f^-1(y) dy = b \beta - a \alpha - \int_a^b f(x) dx$$.

This formula can be seen by drawing an example graph. Along the x-axis, you are integrating f(x)dx. ALong the y-axis you are integrating f-1(y)dy. Draw some rectangles and add/subtract areas to get the formula.