Chop the interval into n strips, add up n rectangles, and let n run off to infinity. It is the long way round, and it is where the integral sign comes from.
Use the limit of a Riemann sum to find the area under y=x from x=0 to x=4
With n strips each is 4 over n wide, and the right edge of the kth is 4k over n.
The sum is 16 over n squared, times the sum of k, which is n times n plus 1 over 2.
As n runs off that goes to 8.
The answer is 8.
Where students go wrong
Letting n go to infinity before summing. The summation formula has to be used first, and only then does the n in it cancel with the n from the width.
Also called limit definition of the integral, area as a limit, Riemann sum limit or summation formulas.
Questions about these worksheets
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Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
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That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the area under a curve by limits questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
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Letting n go to infinity before summing. The summation formula has to be used first, and only then does the n in it cancel with the n from the width.