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Calculus, all content (2017 edition)
Course: Calculus, all content (2017 edition) > Unit 4
Lesson 8: Riemann sumsComparing areas of Riemann sums worksheet
Practice ordering the areas of a left, midpoint, and right Riemann sum from smallest to largest.
Problem 1
Let A denote the area of the shaded region shown below.
We can approximate the exact area A using the following Riemann sums. L, left parenthesis, 6, right parenthesis is the left-hand rule with 6 equal subdivisions. M, left parenthesis, 6, right parenthesis is the midpoint rule with 6 equal subdivisions. R, left parenthesis, 6, right parenthesis is the right-hand rule with 6 equal subdivisions.
Problem 2
Let A denote the area of the shaded region shown below.
We can approximate the exact area A using the following Riemann sums. L, left parenthesis, 6, right parenthesis is the left-hand rule with 6 equal subdivisions. M, left parenthesis, 6, right parenthesis is the midpoint rule with 6 equal subdivisions. R, left parenthesis, 6, right parenthesis is the right-hand rule with 6 equal subdivisions.
Problem 3
Let A denote the area of the shaded region shown below.
We can approximate the exact area A using the following Riemann sums. L, left parenthesis, 6, right parenthesis is the left-hand rule with 6 equal subdivisions. M, left parenthesis, 6, right parenthesis is the midpoint rule with 6 equal subdivisions. R, left parenthesis, 6, right parenthesis is the right-hand rule with 6 equal subdivisions.
Want to join the conversation?
So, it depends on whether the function is increasing or decreasing?(25 votes)
Yes. The area under the curve of a decreasing function will be overestimated by a left-hand Riemann sum and underestimated by a right-hand Riemann sum. The exact opposite is true for an increasing function.(35 votes)
Do the average of the three differents sums ( [L(S) + M(S) + R(S)] / 3 ) would be a better approximation? Or is it just an overkill calculation for an approximate evaluation of the area?(11 votes)
I would think that the M(S) would be the best approximation because it doesn't estimate too much or too less no matter what type of curve. ( [L(S) + M(S) + R(S)] / 3 ) will probably be less accurate than just M(S) because the average contains more deviations from the actual area under curve than M(S) does. I'm not too sure though. It's going to have to depend on the curve.(3 votes)
What if it is both increasing and decreasing?
Does slope, or size of region where it is increasing or decreasing matter?(5 votes)- Yes, those things matter. You would have a much tougher time answering the questions above on a curve with lots of inflection points.(5 votes)
- What happens if its increasing and decreasing(3 votes)
