If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Calculus, all content (2017 edition)

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

Lesson 8: Riemann sums
Math
Calculus, all content (2017 edition)
Integration
Riemann sums
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

Riemann sums worksheet

Google Classroom
Practice identifying and calculating Riemann sums.

Part 1: The three types of Riemann sums

1) What type of Riemann sum is shown below?
Choose 1 answer:

2) What type of Riemann sum is shown below?
Choose 1 answer:

3) What type of Riemann sum is shown below?
Choose 1 answer:

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
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units.

The height is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units.

The area is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
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.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
unitssquared

Part 3: Midpoint Riemann sum

The diagram below shows the midpoint Riemann sum.
Which is an expression for the midpoint Riemann sum?
Choose 1 answer:

Part 4: Right Riemann sum

The diagram below shows the right Riemann sum.
Which is an expression for the right Riemann sum?
Choose 1 answer:

Want to join the conversation?

Log in