Use Simpson's Rule with n=10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator.
x = y + y^(1/2), 1 ≤ y ≤2
there are various online calculators to do this. Just to verify your work, right?
>wink wink<
To estimate the arc length of the curve using Simpson's Rule, we need to follow these steps:
1. Find the first derivative of the function:
Let's find the first derivative of x with respect to y:
dx/dy = 1 + (1/2)y^(-1/2)
2. Calculate the integrand:
Now, calculate the square root of 1 + (dx/dy)^2:
√(1 + (1 + (1/2)y^(-1/2))^2)
3. Determine the interval:
The given interval is 1 ≤ y ≤ 2.
4. Divide the interval into equally spaced subintervals:
Since n=10, we will divide the interval into 10 subintervals of equal width:
Δy = (2 - 1) / 10 = 0.1
5. Estimate the arc length using Simpson's Rule:
Apply Simpson's Rule formula to estimate the integral of the integrand over the interval [1, 2]:
L ≈ Δy/3 * [f(y_0) + 4 * Σf(y_i) + 2 * Σf(y_(i+1)) + f(y_n)]
Plug in the values into the formula:
L ≈ 0.1/3 * [f(1) + 4 * ∑f(y_i) + 2 * ∑f(y_(i+1)) + f(2)]
Now, calculate the approximations for the sums:
∑f(y_i) = f(y_1) + f(y_2) + ... + f(y_9)
∑f(y_(i+1)) = f(y_2) + f(y_3) + ... + f(y_10)
Once you have the values for f(y_i) and f(y_(i+1)), substitute them into the formula to get the final estimate for L.
6. Compare with the calculator's integral:
Use a calculator to evaluate the exact integral of √(1 + (1 + (1/2)y^(-1/2))^2) over the interval [1, 2]. Compare this value with the estimate L obtained using Simpson's Rule.
By following these steps, you should be able to estimate the arc length using Simpson's Rule and compare it with the calculator's integral value.