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Miss Battaglia AB/BC Calculus

4-4 The Fundamental Theorem of Calculus (Day 2) Objective: Evaluate a definite integral using the Fundamental Theorem of Calculus and understand the MVT for integrals. Miss Battaglia AB/BC Calculus. Fundamental Theorem of Calculus. Connects differentiation and integration.

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Miss Battaglia AB/BC Calculus

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  1. 4-4 The Fundamental Theorem of Calculus (Day 2)Objective: Evaluate a definite integral using the Fundamental Theorem of Calculus and understand the MVT for integrals. Miss Battaglia AB/BC Calculus

  2. Fundamental Theorem of Calculus • Connects differentiation and integration. • Integration & differentiation are inverse operations. If a function is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

  3. Guidelines for Using the Fundamental Theorem of Calculus • Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. • When applying the FTC, the following notation is convenient For instance, to evaluate you can write 3. It is not necessary to include a constant of integration C in the antiderivative because

  4. Evaluate each definite integral a. b. c.

  5. A Definite Integral Involving Absolute Value Evaluate

  6. Using the Fundamental Theorem to Find Area Find the area of the region bounded by the graph of y=x3+x, bounded by x=2 and y=0

  7. Thm 4.10 Mean Value Theorem for Integrals If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that

  8. Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a,b], then the average value of f on the interval is

  9. Finding the Average Value of a Function Find the average value of f(x)=9-x2 the interval [-3,3]

  10. The Definite Integral as a Function Evaluate the function at x=0, π/6, π/4, π3, and π/2

  11. The 2nd Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then, for every x in the interval,

  12. Using the 2nd Fundamental Theorem of Calculus Evaluate

  13. Using the 2nd Fundamental Theorem of Calculus Find the derivative of

  14. Thm 4.12 The Net Change Theorem The definite integral of the rate of change of a quantity of F’(x) gives the total change, or net change in that quantity on the interval [a,b].

  15. Using the Net Change Theorem A chemical flows into a storage tank at a rate of 180+3t liters per min, where 0 < t < 60. Find the amount of the chemical that flows into the tank during the first 20 min.

  16. Displacement on [a,b] Total distance traveled on [a,b] =

  17. Solving a Particle Motion Problem A particle is moving along a line so that its velocity is v(t)=t3-10t2+29t-20 ft/sec at time t. • What is the displacement of the particle on the time interval 1 < t < 5? • What is the total distance traveled by the particle on the time interval 1 < t < 5?

  18. Classwork/Homework • Page 294 #73,74,81-86,87,89,91

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