General Contents
Detailed Contents
Index
Programmed tutorial: Integral of the Absolute Value of a Linear
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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.
Additional example at
Additional example.
Find
the
integral
General Contents
Detailed Contents
Index
First let’s look at a related integral involving the same linear quantity, but without the absolute value signs:
Plot the function (2
x
– 3) and show the area involved in this integral.
Then check your work by clicking “Next”.
alt="Graph of a linear factor. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
For this integral, is all of the area positive?
No.
What part is negative?
The part below the
x
-axis, shown in red.
Next plot the function |2
x
– 3| and show the area involved in the original integral
.
Then check your work by clicking “Next”.
alt="Graph of the absolute value of a linear factor. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Is any of the area negative?
No, because of the absolute value signs.
How do we set up the integral?
We handle the integral in two parts, separated by the
x
-intercept.
Determine the
x
-intercept.
We solve for the
x
value which results in a
y
-coordinate of 0:
Set up the integral for the part to the left of the intercept.
Here we
have
, and we get
Do the integration.
Substitute the limits.
We get
What does this represent?
The area of the left triangle.
Set up the integral for the part to the right of the intercept.
Here we
have
, and we get
Do the integration.
Substitute the limits.
What does this represent?
It is the area of the right-hand triangle.
Have we solved the original problem?
Not quite.
We need the integral from -1 to 5.
How can we find that now?
We can add the areas of the two triangles.
Do that.
We get
for the value of the original definite integral.
How can we check our result?
We can determine the area of these two triangles geometrically.
What is the equation for the area of a triangle?
Area = ˝ * base * height
For the left triangle, what is the length of the base?
The distance from
x
= -1 to the intercept at
What is the height of the left triangle?
From the diagram, we see that the height is 5 units.
Use these values to determine the area of the triangle on the left.
In a similar way, determine the area of the right-hand triangle.
Do these geometric results check?
Yes, each agrees with the result of the corresponding integration.
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