Definite Integral: Quadratic over Fractional Power

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Find .

First let's look at this problem graphically.
Draw the graph of this function
in the interval .

alt=" Graph of a quadratic over a fractional power. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag! . .

Shade the area involved in the integral.

alt=" Graph of a fractional power with area shaded. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag! . .

Is this area positive or negative?

Since it is above the x-axis, it is positive.

Now let's start the integration. How do we need to rewrite the problem in order to integrate?

We need to convert it to powers in the numerator and use the Power Rule of integration.

What powers will we get?

We'll get .

Set up the problem with these powers.

We get

Simplify.

Find the indefinite integral I1.

Simplify this expression.

Use this result to set up our definite integral.

Substitute the limits to get I.

We get

Simplify.

We get

Now let's estimate the area graphically. Add a horizontal line to the diagram which omits some of the desired green area and includes some of the white area not involved in the integral.

alt=" Graph of a fractional power showing the average value. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag! . .

What are the dimensions of the rectangle under this line?

It is 8 units wide and 5 units high.

Determine its area.

The area is 8 * 5 = 40.

This area is in approximate agreement with the area in our integration indicating that our integration may be correct.

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

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Index