5.2/3 Definite Integral and the Fundamental Theorem of Calculus Wed Jan 20 Do Now 1)Find the area under f(x) = 3 – x in the interval [0,3] using 3 leftendpt rectangles 2)Integrate f(x) = 3 - x

The Definite Integral When working with Riemann Sums, the width of each rectangle does not have to be uniform, but the idea still persists: Exact area under curve =

The Definite Integral The definite integral of f(x) over [a,b], denoted by the integral sign, is the limit of Riemann sums: When this limit, exists, f(x) is integrable over [a,b]

Theorem If f(x) is continuous on [a, b], or if f(x) is continuous with at most finitely many jump discontinuities, then f(x) is integrable over [a, b]

Area above and below x-axis Any area can be bounded by the x-axis and the function: – If the area is above the x-axis, then it is considered positive – If the area is below the x-axis, then it is considered negative

Ex Calculate

Integral of a Constant For any constant C,

Properties of Integrals

Reversing the Limits of Integration If we reverse the limits of integration,

Additivity for Adjacent Intervals Let, and assume that f(x) is integrable. Then: This is useful for absolute value or piecewise functions

Ex Evaluate the integral

Closure If HW: p.307 # , 55-61

5.3 Fundamental Theorem of Calculus Part 1 Thurs Jan 21 Do Now Use geometry to compute the area represented by the integral

HW Review p.307 #

The Fundamental Theorem of Calculus Part 1 Assume that f(x) is continuous on [a,b]. If F(x) is an antiderivative of f(x) on [a,b], then F(b) – F(a) is considered to be the total change (net change) or accumulation of the function during the interval [a,b]

Notes about FTC1 Notation: We don’t have to worry about + C with definite integrals, because the C’s cancel There’s a calculator function allowed on the AP exam

Ex Calculate the area under the graph of f(x) = x^3 over [2,4] fnInt(x^3,x,2,4)

Ex Find the area under over the interval [1,3]

Ex Find the area under f(x) = sinx on the intervals [0, pi] and [0, 2pi]

Closure Find the area under the function f(x) = 1/x on the intervals [2,8] and [-10,-4] HW: p. 314 #1-59 odds

5.2/3 Definite Integral and FTC1 SKIP Do Now Evaluate each integral 1) 2)

HW Review: p.314 #1-59 odds

Closure How can we find the exact area under a curve over a given interval? Explain HW: none Quiz Tues