Definite Integral: Mixed Fractional Power

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Find .

Let's explore whether this is an odd or even function.
First, is an odd or even function?

Any odd root (3rd, 5th, 7th, etc.) is an odd function. So the first term in our problem is an odd function.

Now consider the –1 term as a function. Is it odd or even?

We can think of this as , or as involving the zero power, we see that it is an even function.

When we have just an odd function, or just an even function, with integration between symmetric limits (–1 and 1 here) we can do some prediction about the result.
Can we do that here with a combination of odd and even functions?

No.

Next let's look at this problem graphically.
Draw the graph of

in the interval .

alt=" Graph of 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. 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 below the x-axis, it is negative.

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

We need to convert it to a power and use the Power Rule of integration.

What power is involved?

The necessary power is .

Rewrite the problem to use this power.

We get

Find the indefinite integral.

Simplify this expression.

Use this result to set up our definite integral.

Substitute the limits to get I1.

We get

How do we evaluate ?

We can think of it as

Evaluate the sixth power of (–1).

It becomes +1.

Take the one fifth root of that.

We get 1.

Now use these results to evaluate I1.

Now let's estimate the area in the third quadrant graphically. Add a horizontal line to the diagram which omits some of the desired red area and includes a compensating amout of the white area not involved in the third quadrant.

alt=" Graph of 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!

What are the dimensions of this rectangle?

It is one unit wide and 1.8 units high.

Determine its area.

The area is 1.0 * 1.8 = 1.8.

Now consider the rectangular area in the fourth quadrant. What are its dimensions?

It is one unit wide and 0.2 units high.

Determine its area.

The area is 1.0 * 0.2 = 0.2.

What is the total of the two graphical estimates?

We get 1.8 + 0.2 = 2.0. This magnitude is in agreement with the magnitude in our integration.

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