AP Calculus AB · Lesson 4 of 5
Antiderivatives and definite integrals
Treat an antiderivative as a family of functions, and a definite integral as accumulated change.
Antiderivatives
An indefinite integral of a function is the family of its antiderivatives, so it includes a constant. The derivative of x² + 7 and the derivative of x² − 3 are the same, 2x. An initial condition chooses the constant.
The power rule for antiderivatives reverses differentiation: the antiderivative of xⁿ is xⁿ⁺¹/(n + 1) plus a constant, when n ≠ −1. Check an antiderivative by differentiating it.
Definite integrals
A definite integral from a to b of a rate gives the net change in the quantity over that interval. If water enters at r(t) liters per minute from t = 0 to t = 2, the integral of r is liters added, not the final amount, until you add the water that was already there.
Net area counts area below the axis as negative. Total area does not. Read the question before you drop the absolute value or keep the signed result.
Worked example
Water enters a tank at r(t) = 3t² liters per minute for 0 ≤ t ≤ 2. The tank starts with 5 liters and has no outflow. How much water is present at t = 2?
- An antiderivative of 3t² is t³.
- The accumulated inflow is 2³ − 0³ = 8 liters.
- The amount present is 5 + 8 = 13 liters.
Why this works. The integral gives net change. The initial amount is added separately.
Check your understanding
- Include + C for an indefinite integral and determine C from an initial condition.
- Differentiate to check an antiderivative.
- Say whether a definite integral represents net change, total amount, or signed area.