Antiderivatives and Definite Integrals
Lessons in this chapter
About Antiderivatives and Definite Integrals
Antiderivatives and Definite Integrals introduces the third foundational idea of calculus: finding the exact area under a curve.
The Area Problem and The Intuition Behind Net Change introduce the challenge of finding an exact area under a curved graph, and Riemann Sums approximates that area using many thin rectangles.
Definite Integrals and Properties of Definite Integrals define the exact area as the limit of those Riemann sums, along with the rules for combining and manipulating definite integrals.
Using Antiderivatives to Evaluate Definite Integrals, Indefinite Integrals, and Method of Substitution build the antiderivative (the reverse of a derivative) as a practical tool for evaluating integrals, and Fundamental Theorem of Calculus closes the chapter by proving that differentiation and integration are exact inverses of each other.
Ten lessons, each a worksheet worked in the page, run from The Area Problem and Riemann Sums through Method of Substitution to Fundamental Theorem of Calculus. The practice problems are scored as you answer them, so the integration formulas get used rather than only read. Free to try, with no login needed for your first 3.
Frequently asked questions
The Area Problem and The Intuition Behind Net Change motivate the integral as an accumulated total. Riemann Sums approximates that area with rectangles, and Definite Integrals and Properties of Definite Integrals make it exact and give the rules for splitting and reversing an interval. Using Antiderivatives to Evaluate Definite Integrals and Indefinite Integrals reverse differentiation, Method of Substitution and Integration using Long Division and Completing the Square add the main techniques, and Fundamental Theorem of Calculus ties differentiation and integration together. This is Unit 6 of the AP Calculus AB course framework, Integration and Accumulation of Change, and it sits well beyond the Common Core.
The area problem asks for the exact area between a curve and the x-axis, a question that (unlike simple shapes) can’t be answered with a basic geometry formula.
A Riemann sum approximates that area by adding up the areas of many thin rectangles drawn under (or over) the curve.
A definite integral is defined as the limit of a Riemann sum as the number of rectangles grows infinitely large and each one grows infinitely thin, giving the exact area.
An antiderivative reverses differentiation, finding a function whose derivative is the given function. It’s the key tool for evaluating a definite integral without summing infinitely many rectangles.
The Fundamental Theorem of Calculus proves that a definite integral can be evaluated directly using an antiderivative, connecting the two big ideas of calculus (differentiation and integration) as exact inverses.