Practice question · Put in order
Order the steps that build the definite integral out of rectangles.
- Identify the limit of the sums as the definite integral
- Choose a sample point x_i in each subinterval and read off the height f(x_i)
- Let n tend to infinity so the strips become arbitrarily thin
- Form the rectangle areas f(x_i) times delta x and add them up
- Partition the interval from a to b into n subintervals of width delta x
Hints
- Nothing can be summed until the strips and their heights exist.
- The limit is the very last act, not the first.
Show the answer
- Partition the interval from a to b into n subintervals of width delta x
- Choose a sample point x_i in each subinterval and read off the height f(x_i)
- Form the rectangle areas f(x_i) times delta x and add them up
- Let n tend to infinity so the strips become arbitrarily thin
- Identify the limit of the sums as the definite integral
Why
Partition, sample, sum, refine, take the limit. Each finite stage is only an approximation; the integral is what the approximations converge to, which is why the definition needs a limit rather than any single sum.
Practise The Definite Integral
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