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5-4 Fundamental Theorem of Calculus Part 2 • If f is continuous at every point of [a, b] and if F is any antiderivative of f on [a, b], then Example • Set up the equation to find the area of the 4 shaded region. F ( x ) ( x 4 x 4)dx 2 x3 Part 1 • If f is continuous on [a, b], then the function has a derivative at every point x in [a, b], and This means… you actually do nothing… no derivative, not antiderivative…. Just switch variables! Example • Find But what if the bound has a derivative??? Part 1 • If f is continuous on [a, b], then the function has a derivative at ever point x in [a, b], and Example • Find Let u = x2 Example • Find dy/dx Must have the variable as the upper bound so…. • Find dy/dx Need to split so only one variable in each and rewrite so both have the variable as upper bound Example • Find dy/dx Example • Find dy/dx Constructing a Function • Find a function y such that dy/dx = tan x and satisfies the condition f(3) = 5 Definite Integrals on the Calculator • Evaluate Homework • P. 302 10-16, 45, 48, 52