Introduction to Integrals: Problems for "Average Value and Second Fundamental Theorem" 4

Introduction to Integrals Problems for "Average Value and Second Fundamental Theorem" 4

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Problem : Find the average value of f for f (x) = 3x² - 6x on [- 1, 1].

Problem : Find the average value of f for f (x) = sin(x) on [- Π, Π].

Problem : For the function above, find the c on [- Π, Π] such that f (c) = f_avg

sin(x) = 0 at x = 0.

Problem : Use the second fundamental theorem of calculus to find F'.

F(x) =

(9t+9)dt

F'(x) = 9x + 9

Problem : Now integrate the above expression for F and then take the derivative to find F'(x) and verify the result from the previous question.

F(x) =

(9t+9)dt

9t+9

dt =

t²+9t

₂^x

x²+9x

- (18 + 18)

F(x) =

x² +9x - 36

F'(x) = 9x + 9

This verifies the result given by the second fundamental theorem.

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