Integral
In short: Integral measures accumulation: a definite integral gives the net area between a graph and the x-axis over an interval, while an indefinite integral gives every antiderivative.
AP Calculus ABCalculusAP Calculus AB Unit 6
What is Integral?
An integral adds up infinitely many tiny pieces. If a rate is graphed against time, the integral over a time span gives the total amount that built up: gallons that flowed, miles that were driven, dollars that were earned.
The definite form \(\int_a^b f(x)\,dx\) is the net signed area between the curve and the x-axis from \(x = a\) to \(x = b\); region above the axis counts positive and below counts negative. The indefinite form \(\int f(x)\,dx\) has no limits and stands for the whole family \(F(x) + C\) of antiderivatives.
How to spot it
- A story gives a rate (gallons per minute, dollars per hour) and asks for the total over a period.
- The problem shows a graph of a rate and asks for an area under it.
- The symbols with a long S and dx appear, with or without numbers at the top and bottom.
- You are asked for net change, such as the change in a quantity given its derivative.
The method
- Decide whether the problem is definite (limits given) or indefinite (no limits, add \(+C\)).
- Find an antiderivative \(F\) of the function being integrated.
- For a definite integral, compute \(F(b) – F(a)\); for an indefinite one, write \(F(x) + C\).
- Attach units: rate units times input units give the units of the accumulated amount.
Integral: solved example
Problem
Water flows into a tank at a rate of \(r(t) = 2t + 1\) gallons per minute. How many gallons enter during the first 3 minutes?
Solution
The total is \(\int_0^3 (2t + 1)\,dt\). An antiderivative is \(F(t) = t^2 + t\).
Then \(F(3) – F(0) = (9 + 3) – 0 = 12\). The tank receives 12 gallons, matching the area of the trapezoid in the figure.
Try it
Question 1
Evaluate \(\int 8x^3\,dx\).
Show the answer
Raise the exponent and divide: \(\frac{8x^4}{4} + C = 2x^4 + C\).
Question 2
Evaluate \(\int_{-2}^{2} x\,dx\) and explain the answer using area.
Show the answer
\(\left[\frac{x^2}{2}\right]_{-2}^{2} = 2 – 2 = 0\). The triangle above the axis (area 2) cancels the triangle below it (signed area \(-2\)).
Integral vs. Antiderivative
An antiderivative is a function whose derivative is the given one. The integral is the broader idea: with limits it is a single number measuring accumulation, and without limits it is the entire family of antiderivatives. The fundamental theorem connects the two ideas.
Keep going
Lessons
Worksheets and quizzes
- Riemann Sums and the Definite Integral – Worksheet
- Riemann Sums and the Definite Integral – Quiz
- Fundamental Theorem of Calculus and Antiderivatives – Worksheet
- Fundamental Theorem of Calculus and Antiderivatives – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus