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### Course: Integral Calculus>Unit 1

Lesson 5: Defining integrals with Riemann sums

# Worked example: Rewriting definite integral as limit of Riemann sum

Given a definite integral expression, we can write the corresponding limit of a Riemann sum with infinite rectangles.

## Want to join the conversation?

• wait at if we're finding the area under the curve don't we have to make sure the height or cos(x) is positive? Why is there no absolute value around cos(pi+pi/n i) in the sigma?
• It's not necessary. It's only necessary when you want the total are. Since integration is used a lot in physics for displacement and the sort, areas under the x-axis are usually considered negative unless told otherwise.
• What happens when you have an "n"?
• "n" is the number of rectangles in the Reimann sum. It's how finely divided the area under the curve is. As n approaches infinity, the Reimann sum approaches a definite integral.
• How do you do this for left sums?
• for left riemann sums basically we just change the functions used for summation to define the height, instead of ∑ from i=1 to N we write ∑ from i=0 to n-1.
But as we use infinite number of rectangles, there will be no significant difference from the result.
CMIIW
• how can I intregate y=2x ?
• After using the reverse power rule, you get x^2.
• we the approximation of the area of the integral gets better and better and better as it approaches infinity does not the error also get better and better and better as it approaches infinity ?
• Pretty much. As an approximation gets closer to the true value, the error in the approximation gets closer to 0.
• I am sorry, I am confused. Sal wrote the summation notation for the right Riemann sum. What would be the summation notation for the left Riemann sum? because I think that in the exercise previous to this video, they write the left Riemann sum in summation notation as Sal did for the right Riemann sum, and when I tried to do the right Riemann sum as Sal did for this video they use a linear equation to find the x sub i.
• Replace i with i-1. This will have the effect of shifting each line 1 to back, making it equivalent to a left Riemann sum.

``       nlim (  Σ  (cos(π + (π/n)(i-1)) * π/n) )n→∞   i=1``
(1 vote)
• How do we calculate those limits specifically (Without turning into integral and then using an integration technique)?
(1 vote)
• It would be quite hard to find the exact integral unless you find the antiderivative of the function (which is sin(x)), which I assumes falls under "integration techniques."
Hope this helped! :)