- Inverse process to differentiation, which is more mechanical.
- We have to "think backwards": "What function has f(x) as its derivative?"
- An important difference between differentiation and
anti-differentiation is that, whereas the derivative of a
function is unique, there is an infinite family of functions
which have a given function as their derivative.
The functions with a common derivative differ only in a constant.
- We introduce a strange notation for the antiderivatives, this
"integral sign", this "indefinite integral". Components include
the integrand, and the differential (dx, usually):
- The integral of a sum is the sum of the integrals (and similarly for a difference).
- The general method for finding an anti-derivative? Stare at it until a solution comes to you!
- When you've stared, and no solution has come to you, it's time to consider alternative strategies: substitution is one.
- This is actually the chain rule in reverse.
- Steps:
- Look for a composition.
- Identify the inside function as g(x)
- differentiate, and see if you don't get g'(x)dx to pop out, and make life simpler.
- thinking of the integrand f(x)dx as the product of the
height (f(x)) and width (dx) of a rectangle, we think of the
definite integral (read
The integral of f(x)dx from a to b
as the area between the curve or the graph of f(x) and the x-axis.
- This is really a limit of finite-sized rectangles:
where the xi values are chosen as representatives of each interval on which a rectangle is constructed. For example, left endpoints could be used, or right endpoints, or midpoints.
- While this idea was developed as an analogy of area, you need to be aware that the definite integral can be positive or negative.
- This theorem ties together the slope problem (differentiation) and
the area problem (integration), while showing us how to solve the area problem:
- The same techniques we used to calculate antiderivatives (e.g. substitution, staring!) continue to work for the definite integrals.
- One tricky addition is the problem of limits: if you do a substitution, you need to make sure that you remember to change the limits of integration to reflect the new variable. The limits of integration represent the values that the differential (dx, often) is stepping off along the axis of the independent variable (usually x).