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Set up and evaluate a definite integral to find the area of the region under the graph of f(x)=e x

−2; on [0,2]

User Jandot
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To find the area under the curve f(x) = e^(x - 2) from 0 to 2, you will need to integrate this function over the interval [0,2]. In other words, you'll compute the definite integral ∫ from 0 to 2 of e^(x - 2) dx.

Here is a step-by-step guide on how to carry out this calculation:

Step 1: Identify the integral boundaries

The problem specifies that we need to find the area under the curve from x = 0 to x = 2. These values are the lower and upper limits of our integral, respectively.

Step 2: Write down the integral

Given the function f(x) = e^(x - 2), and our lower and upper limits which are 0 and 2, the area under the curve is given by:

∫ from 0 to 2 of e^(x - 2) dx

Step 3: Choose a method to evaluate the integral

Since we are dealing with an exponential function, the integration can be done using the Power Rule for integration. We would consider 'e' as a constant and 'x-2' as a variable, and integrate accordingly.

Step 4: Evaluate the integral

Evaluating this integral gives us a value of 0.8646647167633872.

This result represents the exact area under the curve y = e^(x - 2) from x = 0 to x = 2.

User Pranav Labhe
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