Question

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{0}^{2}\left(e^{-2 x}+2\right) d x $$

Short answer

The definite integral of the function \( e^{-2 x} + 2 \) from 0 to 2 can be evaluated as \(-0.5e^{-4} + 0.5 + 4\), and this result represents the area under the graph of the function within the interval from 0 to 2.

Step-by-step solution

Understand the Function

First, it is crucial to understand the function \(e^{-2 x} + 2\). This is an exponential function and should be visualized in a graphing utility software. Sketch the graph of function \(y = e^{-2 x} + 2\) for \(x\) within the interval [0,2].

Calculate the Integral

Next, use the graphing utility to calculate the integral from 0 to 2. The underlying integration formula that you should refer to is: \(\int e^{ax} dx = (1/a)e^{ax} + C\). It is a definite integral so the constant of integration \((C)\) is not required. Thus, the integral of \(e^{-2 x}\) from [0,2] is \[-0.5e^{-2 x} |_{0}^{2}\], which is \[-0.5e^{-4} + 0.5\] adding the integral of 2 from [0,2] which is 2*(2-0) = 4. Thus the total integral is \[-0.5e^{-4} + 0.5 + 4\].

Final Value Calculation and Area

After the integral calculation, it will finally lead to an evaluation in terms of a numerical result. This is where the area interpretation comes into play. This integral also represents the area under the graph of the function within the interval from 0 to 2. So, this area can be visualized as the shaded area under the curve of the function in the graph on the xy-plane, for x within [0,2]. The definite integral in this case would represent a unique numerical value that is associated with this area.