use the trapezoidal and simpson's rule to approximate the value of the definite integral ∫2,1 ln xdx; n=4
compare your result with the exact value of the integral
To approximate the value of the definite integral ∫2,1 ln(x)dx using the trapezoidal and Simpson's rule with n = 4, follow these steps:
1. **Calculate the step size**: The step size (h) can be found by dividing the interval width (b - a) by the number of subintervals (n). In this case, a = 2 and b = 1. So, h = (2 - 1)/4 = 0.25.
2. **Evaluate the function at each point**: Plug the x-values of each subinterval into the function f(x) = ln(x) and calculate their corresponding y-values. In this case, the x-values would be 2, 1.75, 1.5, 1.25, and 1, and the y-values would be ln(2), ln(1.75), ln(1.5), ln(1.25), and ln(1).
3. **Apply the trapezoidal rule**: The trapezoidal rule approximates the integral by summing the areas of trapezoids that span each subinterval. The formula for the trapezoidal rule is:
Approximate = (h/2) * [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)]
Using the values from step 2, the approximation would be:
Approximate = (0.25/2) * [ln(2) + 2ln(1.75) + 2ln(1.5) + 2ln(1.25) + ln(1)]
4. **Apply Simpson's rule**: Simpson's rule provides a more accurate approximation by fitting a parabolic curve through three points and summing their areas. The formula for Simpson's rule is:
Approximate = (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + f(x4)]
Using the values from step 2, the approximation would be:
Approximate = (0.25/3) * [ln(2) + 4ln(1.75) + 2ln(1.5) + 4ln(1.25) + ln(1)]
5. **Compare with the exact value**: To find the exact value of the integral ∫2,1 ln(x)dx, you can use the antiderivative of ln(x), which is x*ln(x) - x. Evaluate this antiderivative from 2 to 1 and subtract the value at 1 from the value at 2.
6. **Compare the results**: Calculate the value of the definite integral using step 5 and compare it with the approximations obtained using the trapezoidal and Simpson's rule in step 3 and step 4, respectively.