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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

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 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.
The fourth rectangle:
The base is
units.

The height is
units.

The area is
unitssquared.

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 parenthesis, x, right parenthesis, equals, 1, plus, 0, point, 1, x, squared on the interval open bracket, 0, comma, 8, close bracket.
unitssquared

## Part 3: Midpoint Riemann sum

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