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Rizzi – Calc BC Integrals represent an accumulated rate of change over an interval The gorilla started at 150 meters The accumulated rate of change was 55 meters Final position was 95 meters In other words: = = − = We can write this in another way: = = = = = = The fundamental theorem of calculus looks at accumulated rates of change: + = () = − () Evaluate the integral: 2 ( 2 − 3) 1 2 − 3 Evaluate the integral: 4 3 1 14 What did the MVT tell us? How is it represented graphically? The MVT for Integrals says: somewhere in the interval [a, b] there is a f(c) value that accurately approximates the area of the curve under the interval. = ()( − ) Find the value of c guaranteed by the Mean Value Theorem for Integrals over the given interval 2 = , [0, 6] 4 =2 3 Rizzi – Calc BC The MVT for Integrals says: somewhere in the interval [a, b] there is a f(c) value that accurately approximates the area of the curve under the interval. = ()( − ) You will NEED this for the AP exam Average value determines the average y-value for a function Average Value Formula: 1 = () − MVT: = ()( − ) Find the average value of = 3 2 − 2 on the interval [1, 4]. Average Value Formula: 1 = () − The derivative of the integral of f(x) is f(x) But why? Find F’(x) = 1 2 2 +1 But what about this? 3 = cos 2 Essentially the same as FTC #1 A chemical flows into a storage tank at a rate of 180 + 3t liters per minute, where 0 ≤ t ≤ 60. Find the amount of the chemical that flows into the tank during the first 20 minutes. 4200 liters When calculating the total distance traveled by the particle, consider the intervals where v(t) ≤ 0 and the intervals where v(t) ≥ 0. When v(t) ≤ 0, the particle moves to the left, and when v(t) ≥ 0, the particle moves to the right. To calculate the total distance traveled, integrate the absolute value of velocity |v(t)|. So, the displacement of a particle and the total distance traveled by a particle over [a, b] is and the total distance traveled by the particle on [a, b] is The velocity (in feet per second) of a particle moving along a line is v(t) = t3 – 10t2 + 29t – 20 where t is the time in seconds. a. What is the displacement of the particle on the time interval 1 ≤ t ≤ 5? b. What is the total distance traveled by the particle on the time interval 1 ≤ t ≤ 5?