Mathematics, five units · Grade 12
What is a definite integral and how do you calculate it?
Integration is differentiation run backwards: you look for the function whose derivative is the one given. A definite integral is that antiderivative evaluated at the upper limit minus its value at the lower limit, and the number it produces is the signed area between the graph and the horizontal axis.
Learning objectives
- Write the antiderivative of a sine, a cosine and an exponential expression
- Evaluate a definite integral between two given limits
- Calculate the area between a wave and the horizontal axis over one arch
- Handle an area that lies partly above and partly below the axis
Integration is differentiation backwards
To integrate a power, raise the exponent by one and divide by the new exponent — the power rule reversed. Every integration rule at this level is a differentiation rule read the other way, which is also how to check one.
An indefinite integral carries a constant, because differentiating destroys constants and nothing can recover which one was there. In a definite integral that constant cancels in the subtraction, which is why it never appears in the final number.
Checking an integration is therefore free: differentiate your answer and see whether you get back the function you started with.
The calculation
Find an antiderivative, substitute the upper limit, substitute the lower limit, and subtract in that order. Reversing the subtraction flips the sign of the whole answer.
The result is a signed area. A region below the horizontal axis contributes a negative amount, so an integral over an interval where the graph crosses the axis is not the same as the total area there.
Area between two graphs
Integrate the upper function minus the lower one, between their points of intersection. Finding those intersections first is part of the question and not a preliminary.
If the two swap over — one is above on part of the interval and below on the rest — split the integral at each crossing. Ignoring a swap lets the two regions cancel and returns a number that is not an area.
Worked examples
Find an antiderivative of f(x) = 2x
- Raise the exponent by one: x squared
- Divide by the new exponent: 2 over 2, which is 1
- Check by differentiating: x squared gives 2x
Answer: x squared, plus a constant
Calculate the definite integral of 2x from 0 to 3
- An antiderivative is x squared
- Substitute the upper limit: 9
- Substitute the lower limit: 0
- Subtract: 9 minus 0
Answer: 9
Calculate the definite integral of x squared from 1 to 2
- An antiderivative is x cubed over 3
- Upper limit: 8 over 3
- Lower limit: 1 over 3
- Subtract: 8 over 3 minus 1 over 3
Answer: 7 over 3
Common mistakes
- Subtracting the limits the wrong way round
- Upper minus lower, in that order. Reversing it flips the sign of the entire answer, which in an area question gives a negative area.
- Treating a signed integral as a total area
- Regions below the axis count negatively. Where the graph crosses the axis, split the integral at the crossing and add the sizes, or the two parts cancel.
- Integrating the lower function minus the upper one
- The order decides the sign. Establish which function is on top over the interval before integrating, rather than assuming from the way they are written.
What to remember
- Integration is differentiation reversed — check by differentiating.
- Upper limit minus lower, in that order.
- The constant cancels in a definite integral.
- Area between graphs: upper minus lower, split at crossings.
More in Mathematics, five units
- How do you prove a trigonometric identity?
- How do you solve a trigonometric equation over a full period?
- How do you find the asymptotes of a rational function?
- How do you differentiate a root and an exponential function?
- What is the natural logarithm and how do you differentiate it?
- How do you find the length of a vector and the angle between two vectors?
- What is the normal distribution and how do you standardise a measurement?