use Simpson's Rule with n=10 to approximate the area of the surface obtained by rotating the curve about the x-axis. y = xln(x) from 1 < x < 2
To approximate the area of the surface obtained by rotating the curve y = x * ln(x) about the x-axis using Simpson's Rule, we need to follow the steps below:
Step 1: Divide the interval [1, 2] into 10 subintervals.
To do this, start by calculating the width of each subinterval:
width = (2 - 1) / 10 = 0.1
Next, create the set of 10 equally spaced x-values within the interval [1, 2]:
x0 = 1 + 0 * 0.1 = 1
x1 = 1 + 1 * 0.1 = 1.1
x2 = 1 + 2 * 0.1 = 1.2
...
x9 = 1 + 9 * 0.1 = 1.9
x10 = 1 + 10 * 0.1 = 2
Step 2: Evaluate the function at each x-value.
For each x-value, compute the corresponding y-value using the function y = x * ln(x):
y0 = x0 * ln(x0)
y1 = x1 * ln(x1)
y2 = x2 * ln(x2)
...
y9 = x9 * ln(x9)
y10 = x10 * ln(x10)
Step 3: Apply Simpson's Rule formula.
The formula for Simpson's Rule is:
integral ≈ h / 3 * [y0 + 4 * (y1 + y3 + y5 + y7 + y9) + 2 * (y2 + y4 + y6 + y8) + y10]
In this case, the integral represents the desired area, and h is the width of each subinterval we calculated in Step 1.
Substituting the values we obtained in Step 2, we can now calculate the approximate area:
approximate area ≈ 0.1 / 3 * [y0 + 4 * (y1 + y3 + y5 + y7 + y9) + 2 * (y2 + y4 + y6 + y8) + y10]
Simply substitute the respective values for y0 to y10 into the formula above and compute the numerical value to obtain the approximate area.
To approximate the area of the surface obtained by rotating the curve y = xln(x) about the x-axis using Simpson's Rule with n = 10, follow these steps:
1. Find the interval width (h) using the formula h = (b - a) / n, where n is the number of subintervals and a = 1 is the lower limit while b = 2 is the upper limit. Calculate h as follows:
h = (2 - 1) / 10
h = 0.1
2. Set up the Simpson's Rule formula:
∫[a,b] f(x) dx ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)]
3. Calculate the values of x and f(x) for each subinterval. In this case, we need to evaluate xln(x) for each x value in the range [1, 2] at intervals of 0.1:
x0 = 1, f(x0) = 1 * ln(1) = 0
x1 = 1.1, f(x1) = 1.1 * ln(1.1) ≈ 0.095
x2 = 1.2, f(x2) = 1.2 * ln(1.2) ≈ 0.182
...
xn-1 = 1.9, f(xn-1) = 1.9 * ln(1.9) ≈ 1.445
xn = 2, f(xn) = 2 * ln(2) ≈ 1.386
4. Substitute the values into the Simpson's Rule formula and calculate the approximation:
Area ≈ (0.1/3) * [0 + 4(0.095) + 2(0.182) + ... + 2(1.445) + 4(1.386) + 1.386]
5. Perform the calculations to find the approximate area.
Note: The numerical integration method may give an approximation, and the accuracy of the result can be improved by increasing the number of subintervals (n).