Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Use the table of integrals to find the exact area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and approximate the area. $$ y=\frac{2}{1+e^{4 x}}, y=0, x=0, x=1 $$

Step-by-Step Solution

Answer

The exact area of the region bounded by the given function and the x-axis between x=0 and x=1 is \( \frac{1-e^{-4}}{2} \)

Step 1: Function Analysis

Given the function \( y=\frac{2}{1+e^{4 x}} \) and the limits \( y=0, x=0,\) and \( x=1 \), the integral of y from x=0 to x=1 needs to be found. This will give the exact area under the graph between these points.

Step 2: Integration of the Function

To find the integral, the exponential integral formula needs to be utilized, which states that the integral of \( e^ax \) dx = \( \frac{e^ax}{a} \). In this case, the fraction with exponential in the denominator can be rewritten as \( 2e^{-4x} \). Hence, \[ \int_{0}^{1} y dx = \int_{0}^{1} 2e^{-4x} dx \] which, using above formula, results in \( -\frac{2}{4}e^{-4x} \big|_0^1 \)

Step 3: Evaluation of the Integral

Evaluating the integral expression at the upper and lower limits, the result is \[ -\frac{e^{-4}}{2} - \left(-\frac{1}{2}\right) = -\frac{e^{-4}-1}{2} \]

Step 4: Solving for the Area

The region is bounded by y=0 and the x-axis, thus, the integral represents the exact area of this region. Even if the result was a negative value, the area is a scalar quantity and is always considered positive. Hence, the exact area is \[ A = \left| -\frac{e^{-4}-1}{2} \right| = \frac{1-e^{-4}}{2} \]