1. Lesson 5-2 The Definite Integral

2. Ice Breaker Find area between x-axis and y = 2x on [0,3] using 3 rectangles and right-end points v t 5 10 0 0 Area of Rectangle = h ∙ w R 1 = (2(x 1 )) ∙ (∆x) = 2(1) ∙ 1 = 2 x i = a + i∆x RH w = ∆x = (b-a)/n = (3-0) / 3 = 1 h = f(x i ) = 2(x i ) RH 1 (2+4+6) = 14 R 2 = (2(x 2 )) ∙ (∆x) = 2(2) ∙ 1 = 4 R 3 = (2(x 3 )) ∙ (∆x) = 2(3) ∙ 1 = 6 Area of Triangle = ½ (b)(h) = ½ (3)(6) = 9

3. Riemann Sums Let f be a function that is defined on the closed interval [a,b]. If ∆ is a partition of [a,b] and ∆x i is the width of the ith interval, c i, is any point in the subinterval, then the sum f(c i )∆x i is called a Riemann Sum of f. Furthermore, if exists, lim f(c i )∆x i we say f is integrable on [a,b]. The definite integral, f(x)dx, is the area under the curve ∑ i=1 n n→∞ ∑ i=1 n ∫ b a

4. Definite Integral vs Riemann Sum Area = f(x) dx ∫ b a Area = Lim ∑A i = Lim ∑f(x i ) ∆x n→∞ i=1 i=n i=1 i=n ∆x = (b – a) / n Area = (3x – 8) dx ∫ 5 2 3i 3 Area = Lim ∑ [3(2 + -----) – 8] (---) n n→∞ i=1 i=n b-a 5-2=3 ∆x [f(x i )] xixi a LH endpoints x i = a + (i-1)∆x RH endpoints x i = a + i∆x

5. Summary & Homework Summary: –Riemann Sums are Limits of Infinite sums –Riemann Sums give exact areas under the curve –Riemann Sums are the definite integral Homework: –pg 390 - 393: 3, 5, 9, 17, 20, 33, 38