Presentation on theme: "Section 5.2: Definite Integrals"— Presentation transcript:
1 Section 5.2: Definite Integrals Objectives:Define a Riemann SumConnect Riemann Sum and Definite IntegralRelate the Definite Integral and Area under the curve
2 Sigma Notationk tells us where to begin, n tells us where to endIf n is ∞, terms go on forever, and ever, and ever, and ever……
3 Reimann SumWe can use sigma notation to approximate the area under a curveWe will add up all the areas of the tiny, little rectangles. We call this a Reimann SumRectangles can lie either above or below the x-axis
4 The Definite Integral as a Limit of Riemann Sums f(x) is on a closed interval [a,b]f is integrable on [a,b] and is the definite integral of f over [a,b]NOTES:is called the partition, and is the longest subinterval length (also may see written as )is the height of the rectangle (it is the value of the function at some value c in the kth subintervalis the width of the rectangle.
5 The Definite Integral of a Continuous Function of [a, b] Let f be continuous [a, b] be partitioned into n subintervals of equal length Δx = (b – a)/n. Then the definite integral of f over [a, b] is given by where each ck is chosen arbitrarily in the kth subinterval. (the more subintervals you have, the more accurate the area)
6 The Existence of Definite Integrals All continuous functions are integrable. That is, if a function f is continuous on an interval [a, b], then its definite integral over [a, b] exists.
7 Definite Integral notation When you find the value of the integral, you have evaluated the integral.The definite integral is a number!!
8 Let’s break it down….. What does all this mean???? Upper limit of Integrationintegrandx is the variable of integrationIntegral SignLower limit ofintegrationRead as “The integral from a to b of f of x dx”
10 Definite Integral and Area Area Under a Curve (as a Definite Integral) If y = f(x) is nonnegative and integrable over a closed interval [a, b] then the area under the curve y = f(x) from a to be is the integral of f from a to b.