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Problem 2 Medium Difficulty

The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate $ \displaystyle \int_0^2 f(x)\ dx $, where $ f $ is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, and 0.9540, and the same number of subintervals were used in each case.
(a) Which rule produced which estimate?
(b) Between which two approximations does the true value $ \displaystyle \int_0^2 f(x)\ dx $ lie?

Answer

a.$R=0.7811 \quad M=0.8632 \quad T=0.8675 \quad L=0.9540$
b.between Midpoint and Trapezoidal

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Video Transcript

in this problem, we are comparing a different methods of finding a Rheiman. Some specifically we're looking at how we can use area of rectangles under a curve, defined the area, and this is going to become very important because we're now learning the definition of an integral and we have to choose which of these methods is the most accurate when we're finding an area. So the left endpoints, the left endpoints is the numbers it takes from the coordinates. They're the largest, so the left endpoints are going to be the largest estimate. That means l the left endpoints must correspond to the number of 0.9 54 Now the right endpoints take the smallest. So the right endpoints is going to be the smallest estimate. So are the right endpoints will be 0.7811 now gets a little tricky. We have to determine if it's the remaining two points are trapezoidal or midpoint. Now, trapezoidal slightly takes a higher number than the midpoint. So the trapezoidal is going to be a little bit more than the midpoint. So T is 0.8675 Now the midpoint is essentially the one that's left, but it's essentially the smallest and most accurate one that we can get, So the midpoint is 10.8632 So for B were asked which ones are the closest estimates to the actual value of our integral? Well, that's going to be, um, the midpoint and tr trapezoidal. Those take into account a more accurate picture of the rectangles under our curve. So those were going to be the most accurate, um, number for the area that are are integral would find. So I hope this helped you understand a little bit more about the methods to finding the area under a curve and learning now which one is the most accurate when we're talking about integration?