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Math 135 Week 12 Lecture Notes Bittenger 10th

Math 135 Week 12 Lecture Notes Bittenger 10th - 5.3...

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Math 135 Class Notes – Week 12 Bittenger 10 th Ed. 1 5.3 Improper Integrals OBJECTIVE: Determine whether an improper integral is convergent or divergent. Solve applied problems involving improper integrals. Example 1: Find the area of the region under the graph of y = 1 x 2 over the interval [1, ) . Such area’s might not always be a finite number . . . Example 2: Find the area of the region under the graph of y = 1 x over the interval [1, ) . Example 3: Determine whether the following integral is convergent or divergent, and calculate its value if it is convergent. 6 e 3 x dx 0 DEFINITION: f ( x ) dx a = lim b 0 f ( x ) dx a b
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Math 135 Class Notes – Week 12 Bittenger 10 th Ed. 2 Example 4: Evaluate the improper integrals that converge. a. 13 x 2 dx −∞ 1 b. 2 x x 2 + 1 dx −∞ c. x 2 e 3 x dx 0 d. e x e x + 1 ( ) 2 dx −∞ DEFINITION: 1. f x ( ) dx = lim a →−∞ −∞ b f x ( ) dx a b 2. f x ( ) dx = f x ( ) dx + −∞ c −∞ f x ( ) dx c where c can be any real number.
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Math 135 Class Notes – Week 12 Bittenger 10 th Ed. 3 5.4 Probability OBJECTIVE: Verify certain properties of probability density functions. Solve applied problems involving probability density functions. Experimental and Theoretical Probability Example 1: Suppose a single card is drawn from a standard deck of playing cards. Find each of the following probabilities: a. getting a jack b. getting a 7 of diamonds c. getting a club d. getting a red card Example 2: An jar contains 4 red balls, 3 green balls, 5 black balls and 6 blue balls, all the same size and weight. One ball is selected without looking. Find the probability of selecting a. Selecting a red ball. b. Selecting a red ball or a green ball. c. Selecting a ball that is not blue.
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