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 using the left Riemann sum, right Riemann sum, and the trapezoid rule, you need to first understand how these methods work and then apply the formulas accordingly. Here's how you can do it:
1. Left Riemann Sum:
To approximate the integral using the left Riemann sum, you divide the interval into subintervals, calculate the value of the function at the left endpoint of each subinterval, and then sum them up.
In this case, you need to divide the interval [0,1] into 8 equal parts. Each subinterval would have a width of (1-0)/8 = 1/8.
Then, for each subinterval, evaluate the function 2/(1+x^2) at the left endpoint. For example, for the first subinterval [0, 1/8], you evaluate the function at x = 0.
Finally, sum up the values for each subinterval and multiply it by the width of each subinterval.
2. Right Riemann Sum:
Similar to the left Riemann sum, the right Riemann sum approximates the integral by dividing the interval into subintervals and calculating the value of the function at the right endpoint of each subinterval.
For the right Riemann sum, you follow the same steps as in the left Riemann sum. The only difference is that you evaluate the function at the right endpoint of each subinterval.
3. Trapezoid Rule:
The trapezoid rule approximates the integral by dividing the interval into subintervals and treating each subinterval as a trapezoid. The area of each trapezoid is the average of the function values at the endpoints of the subinterval multiplied by the width of the subinterval.
For each subinterval, you calculate the values of the function at both endpoints, and then compute the average. Multiply this average by the width of the subinterval.
To estimate the definite integral using the trapezoid rule, you sum up the areas of all the trapezoids for each subinterval.
Once you have applied these formulas for each method, you will have three different estimates for the definite integral.