What Is a Function Defined by an Integral?
Sometimes a function is not given by an ordinary algebraic formula such as . Instead, the function itself is defined using an integral.
For example, consider:
You may not be able to find an elementary antiderivative for . But that does not prevent you from analyzing . In fact, the Fundamental Theorem of Calculus gives you exactly what you need.
The key idea is simple: an integral-defined function can often be differentiated immediately by looking at the integrand.
The Fundamental Theorem Is the Key
If
then the Fundamental Theorem of Calculus tells us that:
So the derivative of the accumulation function is simply the function being accumulated.
Integral defines the function → Differentiate using FTC → Analyze the resulting expression
This is one of the most important patterns to recognize on AP Calculus questions involving functions defined by integrals.
When the Upper Limit Is More Than Just x
There is one important extension. Suppose the function is defined by:
Now the upper limit is rather than simply . The Chain Rule must be used together with the Fundamental Theorem of Calculus.
In words: evaluate the integrand at the upper limit, then multiply by the derivative of the upper limit.
A Step-by-Step Method for Analyzing the Function
- Identify the integrand. This will usually become the derivative of the function.
- Use the Fundamental Theorem of Calculus. Differentiate the integral without trying to calculate the integral itself.
- Find critical points. Set the derivative equal to zero or identify where the derivative is undefined.
- Make a sign chart. Determine where the derivative is positive or negative.
- Determine increasing and decreasing intervals. Positive derivative means increasing; negative derivative means decreasing.
- Classify local extrema. Use the sign change of the derivative.
- For concavity, differentiate again. The second derivative often comes directly from differentiating the integrand.
Worked Example 1: Finding the Derivative
Let . Find .
Notice what we did not do. We did not try to find an antiderivative of the integrand. The Fundamental Theorem lets us differentiate the function immediately.
Worked Example 2: Finding Critical Points
Let . Determine the critical points and the intervals on which is increasing or decreasing.
Reading the Graph of the Integrand
A particularly important AP Calculus skill is understanding that the graph of the integrand tells you about the behavior of the integral-defined function.
Suppose:
Then:
Therefore:
- If , then is increasing.
- If , then is decreasing.
- If changes from negative to positive, has a local minimum.
- If changes from positive to negative, has a local maximum.
tells you the slope of .
The sign of the integrand tells you whether is rising or falling.
The zeros of the integrand identify possible extrema of .
Worked Example 3: Using a Graph of the Integrand
Suppose and the graph of is positive on , zero at , and negative on . Describe the behavior of .
Concavity: Differentiate One More Time
To analyze concavity, find the second derivative. If:
then differentiating again gives:
So the second derivative is simply the derivative of the integrand.
This means you can often determine the concavity of an integral-defined function directly from the slope of the integrand's graph.
- If , then is concave up.
- If , then is concave down.
- If changes sign, there may be an inflection point.
Worked Example 4: Concavity
Let . Determine where is concave up.
Finding the Actual Value of the Function
Sometimes an AP problem asks for a specific value of an integral-defined function rather than its derivative.
For example:
If you know that , then you can evaluate:
The key idea is that a definite integral represents net accumulated change. If the integrand is positive, area is added. If the integrand is negative, area is subtracted.
Using Signed Area
Suppose the graph of is given instead of a formula. You may be asked to calculate .
Break the interval into pieces and use signed area:
- Area above the -axis contributes positively.
- Area below the -axis contributes negatively.
- The total integral is the sum of those signed areas.
Suppose , and the signed areas between and are known. If the net signed area is , and , find .
Common Mistakes
When asked for , you usually do not need an antiderivative. Apply the Fundamental Theorem of Calculus directly. The integrand is often much easier to work with than the original integral.
If the upper limit is rather than , you must multiply by . For example, is not the complete derivative.
If , then . The integrand becomes the derivative of the accumulation function; it is not automatically the function itself.
Area below the -axis counts as negative in a definite integral. Do not add all geometric areas as positive numbers unless the problem specifically asks for total geometric area.
A Fast AP Calculus Strategy
1. See an integral defining a function? Think FTC.
2. Need the derivative? Replace the integral with its integrand.
3. Upper limit is a function of x? Multiply by its derivative.
4. Need increasing/decreasing? Analyze the sign of the derivative.
5. Need extrema? Look for sign changes in the derivative.
6. Need concavity? Differentiate again.
7. Need the function's value? Use the initial value plus signed accumulated area.
Practice Problems
Let . Find .
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Let . Find .
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Let . At which values of can have local extrema?
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Let . Determine where is concave up.
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