What the Definite Integral Is For
The definite integral is one of the central ideas of AP Calculus. It gives us a way to measure accumulated quantity over an interval. When the function is positive, that accumulated quantity can be interpreted as the area between the graph and the x-axis.
Suppose is continuous on the interval from to . The definite integral represents the net area between the graph of and the x-axis over that interval.
The word net matters. Regions above the x-axis contribute positively, while regions below the x-axis contribute negatively.
From Rectangles to an Integral
The basic idea behind integration is surprisingly simple: approximate a complicated region using many thin rectangles, then make the rectangles narrower and narrower.
If an interval from to is divided into equal subintervals, each rectangle has width
If the height of each rectangle is determined by the function, then an approximate area can be written as a sum of rectangle areas:
As the number of rectangles approaches infinity, the approximation becomes exact. This limiting process is the definite integral.
Riemann Sums
A Riemann sum is the mathematical language for adding up the areas of many thin rectangles. On the AP exam, you may encounter left-endpoint, right-endpoint, or midpoint Riemann sums.
Left-Endpoint Approximation
For a left Riemann sum, the height of each rectangle is determined by the function value at the left endpoint of each subinterval.
Right-Endpoint Approximation
For a right Riemann sum, the function is evaluated at the right endpoint of each subinterval.
Midpoint Approximation
For a midpoint Riemann sum, the function is evaluated at the midpoint of each subinterval.
These sums are approximations. The definite integral is what we obtain when the rectangles become infinitely thin.
Worked Examples
Find .
Evaluate .
Suppose crosses the x-axis at . Find the net area represented by .
A continuous function is given by the following values. Use a trapezoidal approximation with four equal subintervals to estimate .
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects the two major ideas of calculus: differentiation and integration.
If is continuous on , and is an antiderivative of , then:
This theorem is why we do not have to add infinitely many rectangles by hand. Instead, we can find an antiderivative and evaluate it at the endpoints.
Definite Integral Versus Indefinite Integral
It is important not to confuse these two notations.
An indefinite integral represents a family of antiderivatives, so it includes a constant of integration. A definite integral has limits of integration and produces a single numerical value.
For example:
The constant disappears in a definite integral because it cancels when the antiderivative is evaluated at the two endpoints.
Area Under a Curve
When is nonnegative on the interval , the definite integral directly represents geometric area:
But if the graph crosses below the x-axis, the integral becomes a signed area. This distinction appears frequently in AP Calculus free-response questions.
Common Mistakes
A definite integral counts area below the x-axis as negative. If the question asks for total area, split the interval at the x-intercepts and add the positive geometric areas.
A definite integral such as is not the same thing as the indefinite integral . The limits tell you exactly which interval is being accumulated.
For a Riemann sum with equal subintervals, always calculate first. Every rectangle's width depends on the interval length and the number of subintervals.
A finite Riemann sum or trapezoidal sum is generally an approximation. The exact definite integral corresponds to the limiting process as the number of subintervals approaches infinity.
Practice Problems
Evaluate .
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Find the area under from to .
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A function is approximated using a left Riemann sum with equal subintervals on . Write the expression for the approximation.
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Suppose is positive on and negative on . If the geometric areas of the two regions are 8 and 3, respectively, find .
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