Question

Express the signed area between the graph of $y=\frac{1}{x}$and the $x$-axis from $x=0.25$to $x=1$in terms of logarithms.

Short answer

The signed area between$y=\frac{1}{x}$from$x=0.25$to$x=1$in terms of logarithms is,$-\ln \left(0.25\right)$.

Step-by-step solution

Step 1. Given information

$y=\frac{1}{x}$and the $x$-axis from $x=0.25$to$x=1$.

Step 2. Find the signed area.

$$\begin{gathered} {\int }_{0.25}^1\frac{1}{x}dx=[\ln x{]}_{0.25}^1 \\ =\ln 1-\ln (0.25) \\ =0-\ln (0.25) \\ =-\ln (0.25) \end{gathered}$$

Therefore, the signed area is,$-\ln \left(0.25\right)$.