Definite integral
The integral of f from a to b is the signed area between the curve and the x-axis. Areas above the axis count positive, below count negative.
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Analogy
A bank balance: money in is positive, money out is negative, and the total is the net change.
Example
Antiderivative: x^3/3.At 3: 27/3 = 9.At 0: 0.Area = 9 - 0 = 9.
In practice
Gives total distance from speed, total water from a flow rate, or total cost from a rate.
Practice
Flashcards
- What is the integral of x^2 from 0 to 3?9.
Questions people ask
What does a negative definite integral mean?
It means the signed area below the x-axis is larger than that above. The integral of sine from pi to 2 pi is -2: the curve lies below the axis, so the area counts as negative. Take the absolute value when you need actual area.
What are the limits a and b?
They are the start and end of the interval you are adding up over. Swapping them changes the sign of the answer. If a equals b, the interval has no width, so the integral is zero. The bounds go in after the antiderivative is found.