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The equation above is the "Fundamental Theorem of Calculus." Given the rules for antidifferentiation, provided that we can find the function F(x) where
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An application of the FTC is finding the area below a curve on an interval [a, b] when the curve is entirely above the x-axis. Below is the graph of a fairly simple area problem.
To calculate this integral, you would set up the following:
This integral can be easily done without "technology." On the AP exam, students will be expected to evaluate integrals of this type without technology and other, "messier" integrals with calculator assistance.
If some of the curve lies below the x-axis, the definite integral will evaluate "area" below the axis as negative. Thus if you are calculating the area of a region, you must take the abolute value of the definite integral on the regions below the x-axis.
For example, to find the area of the region shaded in purple in the graph below,
we need to break the interval up and take the absolute value of the definite integral for the portion of the graph below the x-axis.
Area Between Two Curves:
In the next example, we look at the application of the FTC in finding the area between two curves:
Your first step in solving this is to recall the rule for finding the area between curves. Given two functions, g(x) and h(x), if g is entirely above h, then the area between the two curves on [a, b] is found by the following integral:
There is nothing particularly difficult about this integral, except for the bounds of integration. The intersection points of the curves are "messy decimals." With the two functions entered into your graphing calculator, find the intersection using the [2nd] CALC feature. As soon as you get each, store these into the calculator's memory using STO=> and ALPHA characters. [It makes sense to use A and B, keeping in line with the integration notation.] Then, using the MATH feature of finding the numerical integral, a decimal approximation for the area of the region is quickly found.
Important Test Tip: On the AP test be sure to indicate to your reader that you found the points of intersection with your grapher and show what variables you are assigning to them. This will save you from having to write the messy decimals more than once.
The FTC and Solids of Revolution:
The FTC is also used when finding the volume of a solid of revolution. [See separate page on Solids of Revolution.] The formula for a volume of a solid revolved about the x-axis is
This formula is directly related to the formula for finding the area of a circular disk where the function f is the radius of the circle, hence is called the disk method.
Thus, using the graph of the square root function from above, the volume would be calculated with the integral
.Again, this can be easily evaluated, both with and without technology.
The washer method is employed when the solid has a hollow section. Since the area of a washer is simply a large circular disk minus a smaller circular disk (sharing a common center point), the formula for the volume of such a solid is:
Solids of Revolution about Axes other than the x-axis:
If the region is rotated around a line parallel to the x-axis, such as the line y = 4, as shown below,
The resulting solid will have a hollow center, but the calculation is still a relatively simple. The greatest difficulty is visualizing the resulting figure and the key is to set up the correct integral. Visualize the cross-sections in terms of larger radius (R) and smaller (r) as they move out from the axis of rotation. If the square root function above is rotated about the line y = 4, then the larger (outer) radius is made up by the line y = 4 minus the line y = 0 (the equation for the x-axis):
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Outer Radius: Inner Radius: Thus, the integral that calculates the volume is:
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| The following images show the cross-section of the solid and a 3D representation of the solid. Hopefully this aids you in visualizing the process. | |
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