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 integral, you can use Riemann sums and the trapezoid rule. Let's go step by step through each method:
(i) Left Riemann Sum:
1. Divide the interval from 0 to 1 into 8 equal subintervals. Each subinterval will have a width of (1-0)/8 = 1/8 = 0.125.
2. Evaluate the function at the left endpoint of each subinterval. For each subinterval, substitute the left endpoint into the function 2/(1+x^2). You will get 8 values.
3. Multiply each function value by the width of the subinterval (0.125).
4. Sum up all the products from step 3 to get an estimate of the integral.
INT from 0 to 1 2/(1+x^2) dx ≈ (0.125)(f(0)) + (0.125)(f(0.125)) + (0.125)(f(0.25)) + ... + (0.125)(f(0.875)) + (0.125)(f(1))
(ii) Right Riemann Sum:
1. Divide the interval from 0 to 1 into 8 equal subintervals. Each subinterval will have a width of (1-0)/8 = 1/8 = 0.125.
2. Evaluate the function at the right endpoint of each subinterval. For each subinterval, substitute the right endpoint into the function 2/(1+x^2). You will get 8 values.
3. Multiply each function value by the width of the subinterval (0.125).
4. Sum up all the products from step 3 to get an estimate of the integral.
INT from 0 to 1 2/(1+x^2) dx ≈ (0.125)(f(0.125)) + (0.125)(f(0.25)) + ... + (0.125)(f(0.875)) + (0.125)(f(1)) + (0.125)(f(1.125))
(iii) Trapezoid Rule:
1. Divide the interval from 0 to 1 into 8 equal subintervals. Each subinterval will have a width of (1-0)/8 = 1/8 = 0.125.
2. Evaluate the function at both endpoints of each subinterval. For each subinterval, substitute the left and right endpoints into the function 2/(1+x^2). You will get 9 values since there are 8 subintervals and one extra endpoint.
3. Multiply the function value at both endpoints of each subinterval by the width of the subinterval (0.125).
4. Sum up all the products from step 3, except for the first and last products multiplied by half of the width (0.125/2 = 0.0625), to get an estimate of the integral.
INT from 0 to 1 2/(1+x^2) dx ≈ (0.0625)(f(0)) + (0.125)(f(0.125)) + (0.125)(f(0.25)) + ... + (0.125)(f(0.875)) + (0.125)(f(1)) + (0.0625)(f(1.125))
Remember that these are just estimates, and the accuracy can be improved by using more subintervals.