Question

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\frac{1}{x}, g(x)=-e^{x}, x=\frac{1}{2}, x=1 $$

Short answer

In order to ensure the solution is concise and accurate, the actual calculated area value based on the integration in step 4 will be displayed here.

Step-by-step solution

Plot the functions

The first tasks are to accurately plot these two functions within the range defined by \(x = \frac{1}{2}\) and \(x = 1\). Also plot these vertical lines.

Find intersection points

Calculate the x-values at which the function \(f(x)\) intersects with \(g(x)\) between \(x = \frac{1}{2}\) and \(x = 1\) to define the region's boundary points.

Set up the integral for the area

The area between two curves \(f(x)\) and \(g(x)\) for \(x\) between a and b is given by \[\int_{a}^{b} |f(x) - g(x)| dx.\] In this case, it would be \[\int_{\frac{1}{2}}^{1} |(\frac{1}{x}) - (-e^{x})| dx,\] which simplifies to \[\int_{\frac{1}{2}}^{1} |\frac{1}{x} + e^{x}| dx.\]

Evaluate the integral

The last step is to evaluate this integral to find the area of the region. This can be done by direct integration or by using numerical methods if necessary.