Area under bijective decreasing function

Question

Let $ f:[2,4]\to[3,5]$ be a bijective decreasing function,then find the value of $\int_{2}^{4}f(t) dt-\int_{3}^{5}f^{-1}(t) dt.$

I am not sure whether $\int_{2}^{4}f(t) dt=\int_{3}^{5}f^{-1}(t) dt$ or not.As $f(t)$ and $f^{-1}(t)$ are inverse of each other,so they must be symmetric about line $y=x$.Will this concept be used here or some other method be used.Can someone guide me through?

Try drawing a single diagram showing the area of both integrals. Now compare with the area of the natural rectangles. — Simon S, Jul 27 '15 at 16:55
Following Simon S's suggestion informally: The area under the graph of $f$ over $[2,4]$ is $\int_2^4 f(x),dx$. The area to the left of the graph of $f$ and to the right of $[3,5]$ on the $y$-axis is $\int_3^5 f^{-1}(x), dx$. The difference will leave you with two rectangles. — David Mitra, Jul 27 '15 at 17:06

score 0 · Answer 1 · answered Jul 30 '15 at 15:30

As $f$ is a bijective decreasing function.Therefore,using basic definition of decreasing functions i.e.if$x_1<x_2\Rightarrow f(x_1)>f(x_2)$.Therefore,$f(2)=5,f(4)=3$

$\int\limits_{2}^{4}f(t)dt-\int\limits_{3}^{5}f^{-1}(t)dt=\int\limits_{2}^{4}f(t)dt+\int\limits_{5}^{3}f^{-1}(t)dt=4\times 3-2\times 5=2$

Using,$\int\limits_{a}^{b}f(t)dt+\int\limits_{f^{-1}(a)}^{f^{-1}(b)}f^{-1}(t)dt=bf^{-1}(b)-af^{-1}(a)$

Area under bijective decreasing function

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