Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{1}{x}, y=x^{3}, x=\frac{1}{2}, x=1 $$

Step-by-Step Solution

Answer

The area can be found by evaluating the definite integral \[ \int_{\frac{1}{2}}^{1} (\frac{1}{x} - x^{3}) dx \]

Step 1: Sketch the region

First, draw the four given graphs on the same coordinate plane. The graphs for \(y=\frac{1}{x}\) and \(y=x^{3}\) are curves, whereas the lines \(x=\frac{1}{2}\) and \(x=1\) are vertical lines intersecting the x-axis at points \(\frac{1}{2}\) and 1 respectively. Identify the region enclosed by these four graphs.

Step 2: Determine the top and bottom functions

For \(x\) between \(\frac{1}{2}\) and 1, determine which function lies above the other. In this case, \(y=\frac{1}{x}\) is above \(y=x^{3}\)

Step 3: Calculate the area

Calculate the area of the region as the definite integral from \(\frac{1}{2}\) to 1 of \(y=\frac{1}{x}\) minus \(y=x^{3}\). In other words, the area \(A\) is given by \[A= \int_{\frac{1}{2}}^{1} (\frac{1}{x} - x^{3}) dx \]. Solve this integral to find the area of the region.