Connecting Integral & Differential Calculus

Understanding the fundamental theorem of calculus, anti‐derivatives, indefinite integrals, and mean value theorem for integration and connecting integral and differential calculus.

Mapped to AP College Board # FUN-5, FUN-5.A, FUN-5.A.1, FUN-5.A.2, FUN-5.A.3, FUN-6, FUN-6.B, FUN-6.B.1, FUN-6.B.2, FUN-6.B.3

The Fundamental Theorem of Calculus connects differentiation and integration. Represent accumulation functions using definite integrals. The definite integral can be used to define new functions. If f is a continuous function on an interval containing a, then in the interval [a, b], d/dx [∫f(t)dt] = f(x)] , where x is in the interval. Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as g(x) = ∫f(t)dt over limits a to x. Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration. Evaluate definite integrals analytically using the Fundamental Theorem of Calculus. An antiderivative of a function f is a function g whose derivative is f. If a function f is continuous on an interval containing a, the function defined by F(x) = ∫f(t)dt over limits a to x is an antiderivative of f for x in the interval. If f is continuous on the interval [a, b] and F is an antiderivative of f, then ∫f(x)dx over the limits a to b = F(b)− F(a).