Area between Curves: Lines forming a Triangle

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The points P(-4,4), Q(2,5), and R(8,2) define a triangle. Find its area by integration.

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

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Plot point Q. Then check your work by clicking “Next”.

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Plot point R. Then check your work by clicking “Next”.

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Connect the points. Then check by clicking “Next”.

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Also shown in this diagram is a differential rectangle to be used in finding the area. What is its width?

dx.

Describe which line, PQ, QR, or PR
determines the top of this rectangle.

The line PQ determines the top.

Which line determines the bottom?

PR.

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 triangle by integration.

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

No.

Why?

For rectangles to the right of x = 2, the top of the rectangle is defined by the line QR, as shown in this diagram. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

When we integrate, we’ll need the equation of each line.
Determine the equation of line PQ.

PQ:

Determine the equation of line QR.

QR:

Determine the equation of line PR.

PQ:

Set up the integral for .

Collect like terms in this expression.

Evaluate the indefinite integral.

Substitute the limits.

We get

Set up the integral for .

Collect like terms in this expression.

Evaluate the indefinite integral.

Substitute the limits.

We get

How do we get the total area?

We add these two results to get an area of 36.

Check this result by counting squares in the diagram.

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