# 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.

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).