Question

Estimate the length of the curve y = ln x , 2 ≤ x ≤ 4 , using Simpson’s Rule and four subintervals.

Show the integral and the full set up from Simpson’s Rule. Then use a calculator to get a final decimal
answer, rounding to five decimal places.

Answer 1

To estimate the length of the curve y = ln x using Simpson's Rule, divide the interval [2, 4] into four subintervals and use the formula for Simpson's Rule.

Step-by-step explanation:

To estimate the length of the curve y = ln x using Simpson's Rule, we need to divide the interval [2, 4] into four subintervals. The formula for Simpson's Rule is:

length = (h/3) * [f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + f(b)]

where a and b are the initial and final points of the interval, h = (b - a)/n is the step size, and f(x) is the function. Substituting the values, we have:

length = (2/3) * [ln(2) + 4ln(2+h) + 2ln(2+2h) + 4ln(2+3h) + ln(4)]

Using a calculator, we can evaluate this expression to get the estimated length of the curve.