Area between Curves: Two Exponentials

General Contents

Detailed Contents

Index

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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.



Find the area between . .

First, let’s draw a graph on paper. Draw .
Then check your work by clicking “Next”.

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Add to your diagram.
Then check by clicking “Next”.

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Shade the area to be found. Then check your work by clicking “Next”.

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Show a typical infinitesimal rectangle to be used to find the area. Then check by clicking “Next”.

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What is the width of this rectangle?

dx.

Describe the top of the rectangle.

determines the top.

Describe the bottom.

determines the bottom.

How can we use such rectangles to find the area?

We will describe the area of such rectangles and add them up over the whole area by integration.

Is our above description of the top and bottom applicable over the whole area?

Yes.

Set up the indefinite integral for the integration.

Substitute the expressions for the top and the bottom

Add the limits of integration.

Do the integration.

Will we get 0 when we substitute the lower limit of 0 (as we often do)?

No, because is not equal to 0.

What does it equal?

(anything)0 = 1.

Substitute the limits.

We get

Is this a reasonable area based on a visible estimate on the diagram?

Yes. If we count the marked squares on the diagram, the area is about 2.5 squares. The area of one such square is 5 units by 0.25, or 1.25 square units. So 2.5 * 1.25 = 3.1 is our visual estimate, compared to 3.2 from our integral.

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor

General Contents

Detailed Contents

Index