Calculus problem #2186
What is the Riemann sum of the function f ( x ) = x 3 − 6 x is in the interval [ 0 , 6 ] , if we divide it into 3 equal parts and use the midpoint of each interval?
The answer is 198.
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2 solutions
Using the Riemann sum formula we have $$S=\frac{1}{2}\left( f \left(\frac{1}{2}\right)+f(1)+f \left(\frac{3}{2}\right)+f(2)+f \left(\frac{5}{2}\right)+f(3) \right)$$ $$=\frac{1}{2}\left( -\frac{23}{8}-5-\frac{45}{8}-4+\frac{5}{8}+9 \right)$$ $$=-\frac{63}{16}=\boxed{-3.9375}.$$
@Ricky Escobar Post your answer clearly or do it in a sheet of paper, take a picture of it and then post it.
Given our function f ( x ) = x 3 − 6 x , the Riemann sum as taken from the right is thus:
2 1 i = 1 ∑ 6 f ( i / 2 )
2 1 ( f ( 2 1 ) + f ( 1 ) + f ( 2 3 ) + f ( 2 ) + f ( 2 5 ) + f ( 3 ) )
= 2 1 ( − 8 2 3 − 5 − 8 4 5 − 4 + 8 5 + 9 )
= − 1 6 6 3 = − 3 . 9 3 7 5
I made a program that calculated definite integrals using midpoint rectangle approximation. That is how I solved the question.
Just run the code answering 3 midpoints, top bound=6, and bottom bound=3, and it'll give you the answer.
https://repl.it/@PhysicsAndMath/Numerical-integration-Midpoint-Rule