Question

Evaluate the Riemann sum for f() = 1.2 – 2² over the interval (0, 2) using four subintervals, taking the sample points to be left endpoints. L4 Report answers accurate to 3 places. Remember not to round too early in your calculations. Screen Shot 2020-07-23 at 8.57.43 AM Search over the interval (3, 8) using five approximating Estimate the area under the graph of f(x) rectangles and right endpoints. R. Repeat the approximation using left endpoints. L. Report answers accurate to 4 places. Remember not to round too early in your calculations.

Answer 1

Question 1:

The left Riemann sum (also known as the left endpoint approximation) uses the left endpoints of a subinterval: f(x)dx Ax(f(20) + f(x1) + 2f(x2)+...+fXn-2) + f(xn-1)), where b-a Δr = n We have that a = = 0,b=2, n = 4. 2 – 0 1 Therefore, Ac 4 2 4 subintervals of length Ax 1 2 Divide the interval (0, 2) into n 3 0, 1, 2 = b. Now, we just evaluate the function at the left endpoints: 6 f (x0) = f(a) = f(0) 1.2 19 f(x1) = f (?) = 0.95 2 20 1 f(22) = f(1) 0.2 5 3 21 f(x) = f -1.05 2 20 1 Finally, just sum up the above values and multiply by Ax 2 (1.2 + 0.95 + 0.2 – 1.05) = 0.65 Answer: 0.65

Question 2:

Right Hand approximation

The right Riemann sum (also known as the right endpoint approximation) uses the right endpoints of a subinterval: Sº fla)dæ Ax (f(xi) + f(x) + f(as)+...+1(2n-1) + f(xs), where b-a Δα n We have that a - 3,5 = 8, n = 5. 8 – 3 Therefore, Ax = 1. 5 Divide the interval [3, 8] into n = 5 subintervals of length Ax = 1: a = [3, 4] , [4, 5] , [5,6], [6, 7], [7,8] = b. Now, we just evaluate the function at the right endpoints: 1 f(x1) = f (4) 0.125 8 1 f(22) = f(5) 0.111111111111111 9 f(x3) = f (6) 1 10 0.1 1 f(34) = f(7) 0.0909090909090909 11 f (25) = f(0) = f(8) 1 12 0.083333; 3333333 Finally, just sum up the above values and multiply by Ax = 1: 1(0.125 + 0.111111111111111 +0.1 +0.0909090909090909 + 0.0833333333333333) = 0.510353535353535 Answer: 0.510353535353535

Answer rounded to 4 decimal places will be 0.5104

The left Riemann sum (also known as the left endpoint approximation) uses the left endpoints of a f(x)dx Ax (f(x0) + f(x1) + 2 f(x2)+...+f(In-2) + f(xn-1)), where subinterval: b A2 - a n We have that a = = 3,5 = 8, n = 5. 8 – 3 Therefore, AC = 1. 5 Divide the interval [3, 8] into n = 5 subintervals of length Ax = 1: a= [3, 4] , [4, 5] , [5,6], [6, 7], [7,8] = b. Now, we just evaluate the function at the left endpoints: 1 f(x0) = f(a) = f (3) 0.142857142857143 7 1 f(x1) = f(4) 0.125 1 f (x2) = f(5) 0.111111111111111 9 1 f(x3) = f(6) 0.1 10 f (x4) = f(7) 1 11 0.0909090909090909 Finally, just sum up the above values and multiply by Ax = 1: 1(0.142857142857143 + 0.125 + 0.111111111111111 +0.1 +0.0909090909090909) 0.569877344877345 Answer: 0.569877344877345

Answer rounded to 4 decimal places will be 0.5699

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