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### 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

1) What type of Riemann sum is shown below?

2) What type of Riemann sum is shown below?

3) What type of Riemann sum is shown below?

## 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 $2$ units. The height is $f\left(0\right)=1+0.1\cdot {0}^{2}=1$ unit. The area is $2\cdot 1=2$ units${}^{2}$.
The second rectangle: The base is $2$ units. The height is $f\left(2\right)=1+0.1\cdot {2}^{2}=1.4$ units. The area is $2\cdot 1.4=2.8$ units${}^{2}$.
The third rectangle: The base is $2$ units. The height is $f\left(4\right)=1+0.1\cdot {4}^{2}=2.6$ units. The area is $2\cdot 2.6=5.2$ units${}^{2}$.
The fourth rectangle:
The base is
units.

The height is
units.

The area is
units${}^{2}$.

Now add up the areas of the four rectangles to get the left Riemann sum approximation for the area under the graph of the function $f\left(x\right)=1+0.1{x}^{2}$ on the interval $\left[0,8\right]$.
units${}^{2}$

## Part 3: Midpoint Riemann sum

The diagram below shows the midpoint Riemann sum.
Which is an expression for the midpoint Riemann sum?

## Part 4: Right Riemann sum

The diagram below shows the right Riemann sum.
Which is an expression for the right Riemann sum?

## Want to join the conversation?

• Left Riemann sum is f(left)+f(left+base) and so on?
• 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.
• 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?
• They are used to estimate an integral (area under the curve)
• i wnat defination of bonded function ,partition of I=[a,b],upper RIEMANN AND LOWER RIEMANNof partition
• it appears to me that the mid point Riemann sum gives the closest aproxximation
is this observation correct in general??