8767 #63

[MathCat]

Can anyone explain the answer to #63 on 8767?

Let $$f$$ be a continuous, strictly decreasing, real-valued function such that $$\int_0^\infty f(x)dx$$ is finite and $$f(0)=1$$. In terms of $$f^{-1}$$ (the inverse function of $$f$$), $$\int_0^\infty f(x)dx$$ is...
(a) less than $$\int_1^{\infty} f^{-1}(y)dy$$
(b) greater than $$\int_0^{1} f^{-1}(y)dy$$
(c) equal to $$\int_1^{\infty} f^{-1}(y)dy$$
(d) equal to $$\int_0^{1} f^{-1}(y)dy$$
(e) equal to $$\int_0^{\infty} f^{-1}(y)dy$$

The answer is D.

[antoniechan]

just sketch the graph of the function

shade the region of the given integral

rotate your paper and you will see why the answer is D