Use the trapezoidal rule and simpson's rule to approximate the value of the definite integral ∫2,1 ln xdx; n =4
To approximate the value of the definite integral ∫2,1 ln x dx using the Trapezoidal rule and Simpson's rule with n = 4, follow these steps:
1. Calculate the step size, h, by dividing the range of integration (b - a) by the number of subintervals (n). In this case, the range is from 1 to 2, so the step size is h = (2 - 1)/4 = 0.25.
2. Determine the x-values for the subintervals. Starting from the lower limit (a = 1) and incrementing by the step size, we get the following x-values for n = 4:
x0 = 1
x1 = 1.25
x2 = 1.5
x3 = 1.75
x4 = 2
3. Calculate the approximations for the integral using the Trapezoidal rule:
∫2,1 ln x dx ≈ (h/2) * [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)]
Substitute the x-values into the function f(x) = ln(x):
≈ (0.25/2) * [ln(1) + 2ln(1.25) + 2ln(1.5) + 2ln(1.75) + ln(2)]
Simplify the expression to get the approximation.
4. Calculate the approximations for the integral using Simpson's rule:
∫2,1 ln x dx ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + f(x4)]
Substitute the x-values into the function f(x) = ln(x):
≈ (0.25/3) * [ln(1) + 4ln(1.25) + 2ln(1.5) + 4ln(1.75) + ln(2)]
Simplify the expression to get the approximation.
Note: The actual calculations and simplifications of the approximations can be done using a calculator or software.
To approximate the value of the definite integral ∫2,1 ln xdx using the trapezoidal rule and Simpson's rule with n = 4, we first need to divide the interval [2,1] into subintervals.
Step 1: Divide the interval [2,1] into n subintervals
Since n = 4, we need to divide the interval [2,1] into 4 equal subintervals. The width of each subinterval can be calculated as:
h = (b - a) / n
h = (1 - 2) / 4
h = -0.25
The four subintervals will be:
[2, 2 - 0.25]
[2 - 0.25, 2 - 2(0.25)]
[2 - 2(0.25), 2 - 3(0.25)]
[2 - 3(0.25), 1]
The endpoints of the subintervals are: [2, 1.75], [1.75, 1.5], [1.5, 1.25], [1.25, 1].
Step 2: Apply the trapezoidal rule
The trapezoidal rule approximates the area under the curve by summing the areas of trapezoids formed between consecutive data points. The formula for the trapezoidal rule is:
∫a,b f(x)dx ≈ h/2 * [f(a) + 2f(x2) + 2f(x3) + ... + f(b)]
Using the trapezoidal rule, we can calculate the approximation of the integral as:
Approximation using the trapezoidal rule:
∫2,1 ln x dx ≈ h/2 * [ln(2) + 2ln(1.75) + 2ln(1.5) + 2ln(1.25) + ln(1)]
Substituting the values, we have:
∫2,1 ln x dx ≈ -0.25/2 * [ln(2) + 2ln(1.75) + 2ln(1.5) + 2ln(1.25) + ln(1)]
Step 3: Apply Simpson's rule
Simpson's rule approximates the area under the curve using quadratic polynomial functions over each subinterval. The formula for Simpson's rule is:
∫a,b f(x)dx ≈ h/3 * [f(a) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + f(b)]
Using Simpson's rule, we can calculate the approximation of the integral as:
Approximation using Simpson's rule:
∫2,1 ln x dx ≈ h/3 * [ln(2) + 4ln(1.75) + 2ln(1.5) + 4ln(1.25) + ln(1)]
Substituting the values, we have:
∫2,1 ln x dx ≈ -0.25/3 * [ln(2) + 4ln(1.75) + 2ln(1.5) + 4ln(1.25) + ln(1)]
By evaluating the above expression, you will obtain the numerical approximation of the definite integral.