Presentation on theme: "Cameron Clary. Riemann Sums, the Trapezoidal Rule, and Simpson’s Rule are used to find the area of a certain region between or under curves that usually."— Presentation transcript:
Riemann Sums, the Trapezoidal Rule, and Simpson’s Rule are used to find the area of a certain region between or under curves that usually can not be integrated by hand.
Riemann Sums estimate the area under a curve by using the sum of areas of equal width rectangles placed under a curve. The more rectangles you have, the more accurate the estimated area.
Riemann Sums are placed on a closed integral with the formula: The interval is [a,b] and n is the number of rectangles used Is also called Δx and refers to the width of the rectangles Represents the height of the rectangles
There are three types of Riemann Sums: Left Riemann, Right Riemann, and Midpoint Riemann The left, right, and midpoint refer to the corners of the rectangles and how they are placed on the curve in order to estimate the area.
Left Riemann Sums place the left corner of the rectangles used to estimate the area on the curve. Left Riemann sums are an underestimation of the area under a curve due to the empty space between the rectangles and the curve.
Right Riemann sums place the Right corner of the rectangles on the curve. Right Riemann Sums are an overestimation of area because of all the extra space that is not under the curve that is still calculated in the area because it is inside the rectangles
Midpoint Riemann Sums place the middle of the Rectangle on the curve Midpoint Riemann Sums are the most accurate because the area found in the part of the rectangle that is over the curve makes up for the area lost in the space between the curve and the rectangle
First, find the width of the rectangles or Δx On the interval [0,1] with n=4 Then, starting with a, the first number on the interval, plug the numbers into the formula, adding Δx each time. So… *You should always begin with a and end with b, if not, you plugged in the numbers wrong
Once you have the numbers and You can plug them into the formula : When doing a Left Riemann, plug all numbers into the formula except for the last number. When doing a Right Riemann, plug in all numbers into the formula except for the first number. When doing a Midpoint Riemann, average the numbers and then plug in those values to the formula.
If the area can be found by hand, you can compare your answers from the Riemann Sums to the actual answer to see how accurate your estimation was In this problem, we can find the area by hand. The area for this problem is: Comparing the answers, the area found using the Left Riemann was under the amount of the actual area, the Right Riemann was over the amount of the actual area, and the Midpoint Riemann was the closest to the actual answer. None of the Riemann Sum types gave the exact answer, but that is because they are estimations.
Calculate the Left and Right Riemann Sum for on [0, π] using 4 rectangles.
Trapezoidal Rule is very similar to the Riemann Sums, but instead of using rectangles to approximate area, it uses trapezoids. The trapezoidal rule is more accurate than the Riemann sums.
When using the Trapezoidal Rule, use the formula: The reason all but the first and last functions are multiplied by two is because their sides are shared by two trapezoids. Is still added to each like in the Riemann Sums
Use the Trapezoidal Rule to Calculate on the interval [0,1] when n=4 Δx= Once you have all this information, all you have to do is plug the numbers into the formula
Just like in the Riemann Sums, if the area can be found by hand, you can sue that answer to check to see how close the estimate was to the exact answer. In this particular problem, the exact answer is 1/3units squared or.3333 units squared. Using the Trapezoidal Rule, the estimate comes out to be.34375 units squared. The estimated answer is very close to the exact answer.
Calculate the Trapezoidal Rule for on the interval [1,2] when n=5
1. Find the Left Riemann of on [0,2] when n=6 2. Find the Right Riemann of on [0,2] when n=6 3. Find the Midpoint Riemann of on [0,2] when n=6 4. Calculate the Trapezoidal rule for 5. Calculate the Simpson’s Rule for on [0,π] for n=4 on [2,4] where n=4
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