Unit 6 is the heart of the second half of AP Calculus. In this unit, we transition from the "slope of a curve" (Derivatives) to the "area under a curve" (Integrals). Having taught this for over 20 years, I’ve found that students who master the Fundamental Theorem of Calculus here are the ones who consistently score 4s and 5s in May.
Definite Integrals as Accumulation: Understanding how a rate of change accumulates into a net change.
Riemann Sums: Approximating area using Left, Right, Midpoint, and Trapezoidal sums.
The Fundamental Theorem of Calculus (FTC): Connecting differentiation and integration as inverse processes.
Integration Techniques: Mastering u-substitution and finding antiderivatives of basic and transcendental functions.
"The most common mistake I see students make in Unit 6 is forgetting the Constant of Integration (+C) on indefinite integrals. On the AP Free Response Questions (FRQs), forgetting +C can cost you. Another tip: Always look for the 'inner function' when deciding if you need u-substitution!"
Answer: The primary difference is the result and the notation. An Indefinite Integral is the general family of antiderivatives for a function; it always requires a +C (constant of integration) at the end. A Definite Integral has specific upper and lower bounds (numbers on the integral sign) and results in a single numerical value representing the net accumulation or area under the curve.
Answer: In my 20 years of teaching, this is one of the most common places students lose points! You include +C only when finding an Indefinite Integral (the antiderivative). If the integral has "bounds" (numbers at the top and bottom of the integral symbol), it is a Definite Integral, and the constants cancel out, so no +C is needed.
Answer: Think of u-substitution as the "Reverse Chain Rule." Look for a "composite function" where one part of the integrand is the derivative of another part. A great tip is to look for a "trapped" function—something inside a square root, a denominator, or an exponent—whose derivative is sitting right there next to it.
Answer: No, a Riemann Sum (Left, Right, Midpoint, or Trapezoidal) is only an approximation. The approximation becomes "exact" only when we take the limit as the number of sub-intervals (n) approaches infinity. On the AP Exam, you’ll often be asked if a Right or Left sum is an overestimate or underestimate based on whether the function is increasing or decreasing.