Introduction to Integrals

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

favg		= f (x)dx
		= x3-3x2-11
		= 1-3-(-1-3)
		= 1

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

favg		= f (x)dx
		= -cos(x)-ΠΠ
		= --1-(-(-1))
		= 0

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

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+9dt = t2+9t2x
x2+9x - (18 + 18)
F(x) = x2 +9x - 36
F'(x) = 9x + 9

This verifies the result given by the second fundamental theorem.