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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition) > Unit 4

Lesson 8: Riemann sums# Riemann sums worksheet

Practice identifying and calculating Riemann sums.

## Part 1: The three types of Riemann sums

## Part 2: Left Riemann sum

The diagram below shows the left Riemann sum. We want to find the total area of the four rectangles.

**The first rectangle:**The base is start color #11accd, 2, end color #11accd units. The height is f, left parenthesis, 0, right parenthesis, equals, 1, plus, 0, point, 1, dot, 0, squared, equals, start color #1fab54, 1, end color #1fab54 unit. The area is start color #11accd, 2, end color #11accd, dot, start color #1fab54, 1, end color #1fab54, equals, 2 unitssquared.

**The second rectangle:**The base is start color #11accd, 2, end color #11accd units. The height is f, left parenthesis, 2, right parenthesis, equals, 1, plus, 0, point, 1, dot, 2, squared, equals, start color #1fab54, 1, point, 4, end color #1fab54 units. The area is start color #11accd, 2, end color #11accd, dot, start color #1fab54, 1, point, 4, end color #1fab54, equals, 2, point, 8 unitssquared.

**The third rectangle:**The base is start color #11accd, 2, end color #11accd units. The height is f, left parenthesis, 4, right parenthesis, equals, 1, plus, 0, point, 1, dot, 4, squared, equals, start color #1fab54, 2, point, 6, end color #1fab54 units. The area is start color #11accd, 2, end color #11accd, dot, start color #1fab54, 2, point, 6, end color #1fab54, equals, 5, point, 2 unitssquared.

## Part 3: Midpoint Riemann sum

The diagram below shows the midpoint Riemann sum.

## Part 4: Right Riemann sum

The diagram below shows the right Riemann sum.

## Want to join the conversation?

- Left Riemann sum is f(left)+f(left+base) and so on?(12 votes)
- Yes! It appears to be. (I'm assuming you know to multiply the result by delta-x or base)

It would be helpful to have a formal definition of them attached to this worksheet or introduced beforehand. I haven't seen them before this. (I have been following the Integral Calculus course outline.(8 votes)

- Midpoint reimann sum gives a better approximation than left-hand and right-hand.So,what is the point of using left-hand and right-hand reimann sums?(10 votes)
- They are used to estimate an integral (area under the curve)(0 votes)

- i wnat defination of bonded function ,partition of I=[a,b],upper RIEMANN AND LOWER RIEMANNof partition(0 votes)
- it appears to me that the mid point Riemann sum gives the closest aproxximation

is this observation correct in general??(3 votes)- I think, for general functions, it is true that mid point Riemann sum gives better approximation than the side ones when you are using a few rectangles to approximate.

: )(2 votes)

- May you do a video on how to do midpoints?(3 votes)
- I think it's not needed. It literaly same as the graph shown in first Riemann sums video, only the height of rectangles is from bottom to point, where width middle of rectangle intersect the f(x). There is really nothing harder behind that.(2 votes)

- Apparently there's no video about right and midpoint riemann sum. Neither in the progress panel on the left.(1 vote)
- Right Riemann sums are like f(x+1)+f(x+2) instead of f(x)+f(x+1).(1 vote)

- find the area under the graph of f(x)=x+x^2+x^3 over the [0,1] by computing lim as N->infinity Rn how do i start this problem(0 votes)