Area preservation of $xf(x)$ in a logarithmic plot

Question

Say we're interested in a quantity $Q$ which is the area under a function $f(x)$ in a linear plot of $f(x)$ vs. $x$. I heard my professor say that if we plot $xf(x)$ in a semilog x plot, the area under the curve in this plot preserves the quantity $Q$. I cannot see how this can be true. Does the fact the derivative of $\ln(x)$ is $1/x$ play a role?

By "semilog x", I believe she means a plot linear on the y-axis and logarithmic on the x-axis.

Answer 1

Let the endpoints be $a$ and $b$. In the semilog plot, we plot $xf(x) = e^y f(e^y)$ as a function of $y = \log(x)$. The endpoints then are $\log(a)$ and $\log(b)$ and the area under the curve is $$ \int_{\log(a)}^{\log(b)} e^y f(e^y) \, dy = \int_{a}^{b} f(x) \, dx = Q $$ which follows from the substitution rule.

Answer 2

In the x-semilog plot, the area under the curve is $A = \int exp(x)*f(exp(x)) $. The variable change $y = exp(x)$ yields $dx = dy/y$, so $ A = \int y/y*f(y) = \int f(y)$