In this video we talk about when trapezoidal rule overestimates the area under the curve, when it underestimates the area under the curve, and when it finds exact area.
0:38 What does trapezoidal rule do when the curve is CONCAVE DOWN? // In general, when a curve is concave down, trapezoidal rule will underestimate the area, because when you connect the left and right sides of the trapezoid to the curve, and then connect those two points to form the top of the trapezoid, you’ll be left with a small space above the trapezoid. The small space is outside of the trapezoid, but still under the curve, which means that it’ll get missed in the trapezoidal rule estimate, even though it’s part of the area under the curve. Which means that trapezoidal rule will consistently underestimate the area under the curve when the curve is concave down.
1:24 What does trapezoidal rule do when the curve is CONCAVE UP? // The opposite is true when a curve is concave up. In that case, each trapezoid will include a small amount of area that’s above the curve. Since that area is above the curve, but inside the trapezoid, it’ll get included in the trapezoidal rule estimate, even though it shouldn’t be because it’s not part of the area under the curve. Which means that trapezoidal rule will consistently overestimate the area under the curve when the curve is concave up.
2:09 Summary // So if the trapezoidal rule underestimates area when the curve is concave down, and overestimates area when the curve is concave up, then it makes sense that trapezoidal rule would find exact area when the curve is a straight line, or when the function is a linear function.
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Keep On Going by Joakim Karud https://soundcloud.com/joakimkarud
Creative Commons — Attribution-ShareAlike 3.0 Unported— CC BY-SA 3.0
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student—from basic middle school classes to advanced college calculus—figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: http://www.kristakingmath.com
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Tagged under: overestimate,calculus,area curve,underestimate,area estimation,approximating area,trapezoidal rule,exact area,Krista King,educational,estimating area,approximation,area approximation
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