How do you evaluate the definite integral by the limit definition given #int x/2dx# from [0,4]?
2 Answers
4
Explanation:
Please see the explanation section below.
Explanation:
Here is a limit definition of the definite integral. (I hope it's the one you are using.) I will use what I think is somewhat standard notation in US textbooks.
.
Where, for each positive integer
And for
I prefer to do this type of problem one small step at a time.
Find
For each
Find
And
Find
# = 1/2 * (4i)/n#
# = (2i)/n#
Find and simplify
# = sum_(i=1)^n( (8i)/n^2)#
# = 8/n^2 sum_(i=1)^n(i)#
Evaluate the sums
# = 8/n^2((n(n+1))/2)#
(We used a summation formula for the sums in the previous step.)
Rewrite before finding the limit
# = 4((n(n+1))/n^2))#
Now we need to evaluate the limit as
To finish the calculation, we have
# = 4(1) = 4#