Calculus, all content (2017 edition)
Practice identifying and calculating 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 units. The height is unit. The area is units.
The second rectangle: The base is units. The height is units. The area is units.
The third rectangle: The base is units. The height is units. The area is units.
The fourth rectangle:
The base is
The height is
The area is
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 on the interval .
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?(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)
- 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)
- 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)