Aren't They Just The Same?

Calculus Level 3

$A = \int_{0}^{1} f(2x)\,dx \qquad \qquad B = \int_{0}^{2} f(x) \,dx$

If $f(x)$ is always positive, which is bigger?

A

B

They are equal

2 solutions

Rishabh Jain
Jul 12, 2016

For $A$ substitute $2x=t$ such that $2dx=dt$

$\color{#D61F06}{A}=\dfrac 12\int_0^2f(t)dt=\dfrac 12\int_0^2 f(x)dx=\color{#D61F06}{\dfrac{B}2}$

$\Large \therefore B>A$

J D
Jul 12, 2016

Using a+b-x, A comes out to be the integral from 0 to 1 of f(2-2x)dx and B comes out to be the integral from 0 to 2 of f(2-x)dx. f(2-x) will have larger y values than f(2-2x), so integral B is bigger.

Aren't They Just The Same?

2 solutions

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