Can someone tell me how to do these?
Estimate INT from 0 to 1 2/(1+x^2)dx by subdividing the interval into 8 parts, using
(i) the left Riemann sum: INT from 0 to 1 2/(1+x^2)dx
(ii) the right Riemann sum: INT from 0 to 1 2/(1+x^2)dx
(iii) the trapezoid rule INT from 0 to 1 2/(1+x^2)dx
To estimate the definite integral from 0 to 1 of the function f(x) = 2/(1+x^2)dx using different approximation methods, you can follow these steps:
(i) Estimate using the Left Riemann sum:
1. Divide the interval from 0 to 1 into 8 equal parts (subintervals).
2. Calculate the width (Δx) of each subinterval by dividing the total interval width (1 - 0 = 1) by the number of subintervals (8).
Δx = (1 - 0) / 8 = 1/8 = 0.125
3. Evaluate the function at the left endpoint of each subinterval and multiply it by the subinterval width (Δx).
4. Sum up all the individual products to get the left Riemann sum.
(ii) Estimate using the Right Riemann sum:
1. Divide the interval from 0 to 1 into 8 equal parts (subintervals).
2. Calculate the width (Δx) of each subinterval by dividing the total interval width (1 - 0 = 1) by the number of subintervals (8).
Δx = (1 - 0) / 8 = 1/8 = 0.125
3. Evaluate the function at the right endpoint of each subinterval and multiply it by the subinterval width (Δx).
4. Sum up all the individual products to get the right Riemann sum.
(iii) Estimate using the Trapezoid rule:
1. Divide the interval from 0 to 1 into 8 equal parts (subintervals).
2. Calculate the width (Δx) of each subinterval by dividing the total interval width (1 - 0 = 1) by the number of subintervals (8).
Δx = (1 - 0) / 8 = 1/8 = 0.125
3. Evaluate the function at both the left and right endpoints of each subinterval.
4. Multiply the average of the function values at each endpoint by the subinterval width (Δx).
5. Sum up all the individual products to get the estimated integral using the trapezoid rule.
Once you have the estimated values for each method, you may compare them to determine which approximation provides a better estimation for the integral.