5.4 Indefinite Integrals and the Net Change Theorem
An indefinite integral is basically the antiderivative of the function. It doesn't have upper and lower bounds because that would make it a definite integral. Indefinite integrals need the +C!
Table of Indefinite Integrals
Find the general indefinite integral
Using the formula:
THE NET CHANGE THEOREM
The Integral of a rate of change is the net change (displacement for position functions)
Basically this theorem states that the integral of f or F' from a to b is the area between a and b or the difference in area from the postion of F(a) to F(b).
This can be applied to things such as:
So for a velocity function:
To calculate displacement we can use the equation
to calculate total distance traveled we can add the absolute values of the areas of each sector from each x intercept to the next x intercept
A particle moves along a line so that its velocity at time t is
a) find the displacement from t=[1,4]
b) find the distance traveled during that time period
Finding the displacement:
Finding the total distance traveled during that time period
The total distance traveled and the displacement are the same because the position function does not pass below the x axis therefore there are no negative areas. If there were negative areas the displacement would be a smaller number and the distance would stay the same.
A Lesser Lesson in Indefinite Integrals but helpful nontheless
Magnus You're Up NeXT!!!