Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

Get the most by viewing this topic in your current grade. Pick your course now.

?
Intros
Lessons
  1. Riemann Sum overview:
  2. area under the curve
  3. summation formula
  4. what is "Riemann Sum"?
  5. integrals expressed as Riemann Sum
?
Examples
Lessons
  1. Finding the area under the graph of a function using a graphing calculator.
    Consider the function f(x)=x2f\left( x \right) = {x^2}, 1x31 \le x \le 3.
    Find the area under the graph of ff using a graphing calculator.
    1. Finding the area under the graph of a function using the Riemann Sum.
      Consider the function f(x)=x2f\left( x \right) = {x^2}, 1x31 \le x \le 3.
      Estimate the area under the graph of ff using four approximating rectangles and taking the sample points to be:
      1. right endpoints
      2. left endpoints
      3. midpoints
    2. Evaluating integrals with a Riemann Sum
      Consider the function f(x)=x25x+3f\left( x \right) = {x^2} - 5x + 3, 2x52 \le x \le 5.
      1. Evaluate the Right Riemann sum for ff with 6 sub-intervals.
      2. Evaluate 25(x25x+3)dx\int_2^5 \left( {{x^2} - 5x + 3} \right)dx, by finding the Riemann sum for ff with \infty intervals.
    3. Evaluating Riemann Sum with trapezoids
      Consider the function f(x)=x2f\left( x \right) = {x^2}, 1x31 \le x \le 3. Estimate the area under the graph of ff using four approximating trapezoids.
      Free to Join!
      StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
      • Easily See Your Progress

        We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
      • Make Use of Our Learning Aids

        Last Viewed
        Practice Accuracy
        Suggested Tasks

        Get quick access to the topic you're currently learning.

        See how well your practice sessions are going over time.

        Stay on track with our daily recommendations.

      • Earn Achievements as You Learn

        Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
      • Create and Customize Your Avatar

        Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
      Topic Notes
      ?
      In this lesson, we will learn how to approximate the area under the curve using rectangles. First we notice that finding the area under the curve is easy if the function is a straight line. However, what if it is a curve? We will actually have to approximate curves using a method called "Riemann Sum". This method involves finding the length of each sub-interval (delta x), and finding the points of interest, finding the y values of each point of interest, and then use the find the area of each rectangle to sum them up. There are 3 methods in using the Riemann Sum. First is the "Right Riemann Sum", second is the "Left Riemann Sum", and third is the "Middle Riemann Sum". Lastly, we will look at the idea of infinite sub-intervals (which leads to integrals) to exactly calculate the area under the curve.

      definite integral expressed as riemann sum, and summation formula
      Basic Concepts
      ?