5.3 Notes - Definite Integrals (Section 5.3) Recall:...

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Unformatted text preview: Definite Integrals (Section 5.3) Recall: We’ve learned several methods to approximate area under a curve with rectangles. The more rectangles we use, the better the approximation. The exact area is: Definition: If f is a continuous function on [a, b], we divide [a, b] into n subintervals of equal n b b−a width, Δx = , the definite integral of f from a to b is ∫ f ( x ) dx = lim ∑ f ( xi ) Δx a n→ ∞ n i =1 where xi is some point in the ith subinterval. n Alternate Definition: Can also say ∫ f ( x ) dx = lim ∑ f ( xi ) Δxi where P = { x0 , x1 , x2 , ..., xn } of b a P →0 i =1 [a, b] and Δxi = xi − xi −1 . (In this definition, the distance between consecutive xi is not necessarily the same). Vocabulary: n ∑ f ( x )Δx is a _____________________ ___________. i =1 i ∫ is an ____________________ ________________. a is the _________________ ________________ of integration. b is the _________________ ________________ of integration. f ( x ) is the ___________________________. dx tells us the ________________________ of integration. Notes: b 1. When ∫ f ( x ) dx exists, we say that f is ______________________________. a 2. Any _____________________________ function is integrable. b 3. When the graph of f is above the x – axis between a and b, ∫ f ( x ) dx is the area between the a graph of f and the x – axis. b 4. When the graph of f goes below the x – axis, ∫ f ( x ) dx denotes the net area which means a b ∫ f ( x ) dx = _________________________________ ‐ ____________________________________ a Example 1: Write in integral notation: 2 n a) lim ∑1 − ( xi ) Δx [ −2, 1] n→ ∞ i =1 n b) lim ∑ n→ ∞ i =1 ( ) Δx [ 3, 100 ] ln xi * xi * Evaluating Definite Integrals: We have 3 methods right now that we can use to evaluate definite integrals: 1) Estimating using Riemann sums. (not exact) 2) Using the limit definition of the integral. (exact) 3) Using what we know about the area of a particular curve. (exact) 2 Example 2 (Estimating with Riemann Sums): Estimate ∫ x 2 − 1 dx using n = 4 and: 0 a) Left Riemann sum: b) Right Riemann sum: c) Midpoint Riemann sum: Using the limit definition: This method will use our basic formulas for sums of integers: n n(n + 1) ∑k = 2 k =1 n ∑k k =1 2 = n(n + 1)(2 n + 1) 6 ⎛ n(n + 1) ⎞ ∑ k3 = ⎜ 2 ⎟ ⎝ ⎠ k =1 n 2 2 Example 3: Evaluate ∫ x 2 dx . 0 Δx = xi = a + i Δx = this uses right endpoints (using left endpts. gives the same answer) Then 2 n 2 2 ∫ x dx = lim ∑ xi Δx = 0 n→ ∞ i =1 Example 4: Use area to compute the following: 4 a) ∫ 16 − x 2 dx 0 3 b) ∫ ( x − 1 + 1)dx 0 Properties (pg 336): ∫ 2. ∫ 3. ∫ 4. ∫ 5. ∫ 6. ∫ 1. b a a a b a b a b a b a a f ( x )dx = − ∫ f ( x )dx b f ( x )dx = 0 c dx = c ( b − a ) f ( x ) ± g( x ) dx = ∫ b a b f ( x )dx ± ∫ g( x )dx a b c f ( x ) dx = c ∫ f ( x ) dx , c a constant a f ( x )dx = ∫ c a b f ( x )dx + ∫ f ( x )dx c Comparison Properties: 7. (Max‐Min Inequality) If f has a maximum value (max f) and minimum value (min f) on an b interval [ a, b ] , then (min f )(b − a ) ≤ ∫ f ( x ) dx ≤ (max f )(b − a ) a b b a a 8. If f ( x ) ≤ g( x ) on [ a, b ] then ∫ f ( x ) dx ≤ ∫ g( x ) dx Example 5: Suppose the function g whose graph is shown below is even and ∫ 1 0 g( x ) dx = 1.5 ∫ 0 1. ∫ 1.15 1 2. ∫ g( x ) dx 1 3. ∫ 2 −1.15 0 1.15 g( x ) dx 1.15 4. ∫ 2 g( x ) dx = 2 ∫ g( x ) dx = 3 . −2 Find: 1.15 g( x ) dx 6 g( x ) dx Example 6: Suppose g′( x ) (whose graph is given) is an odd function and ∫ 2 −1 2 g′( x ) dx = 2 and ∫ g′( x ) dx = 3 . 0 Find: 1 a. ∫ g′( x ) dx −1 2 b. ∫ g′( x ) dx 1 2 c. ∫ g′( x ) dx 2 0 d . ∫ g′( x ) dx −1 2 e. ∫ − g′( x ) dx 0 a Question: What do these two examples tell us about ∫ f ( x ) dx for: a) f an even function? b) f an odd function? −a ...
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This note was uploaded on 10/02/2011 for the course MATH 2214 at Virginia Tech.

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