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, we need to divide the interval [1, 2] into smaller subintervals and evaluate the function at specific points within each subinterval.
Step 1: Divide the interval [1, 2] into smaller subintervals.
Since n = 4, we will divide the interval into four equal subintervals of width h.
h = (2 - 1)/4 = 1/4 = 0.25
The four subintervals are: [1, 1.25], [1.25, 1.5], [1.5, 1.75], and [1.75, 2].
Step 2: Evaluate the function within each subinterval.
For the trapezoidal rule, we need to evaluate the function at the endpoints of each subinterval.
For the Simpson's rule, we need to evaluate the function at the endpoints and the midpoint of each subinterval.
In this case, the function is ln(x).
Evaluate ln(x) at the endpoints and the midpoint of each subinterval:
- For the trapezoidal rule:
f(1) ≈ ln(1)
f(1.25) ≈ ln(1.25)
f(1.5) ≈ ln(1.5)
f(1.75) ≈ ln(1.75)
f(2) ≈ ln(2)
- For the Simpson's rule:
f(1) ≈ ln(1)
f(1.125) ≈ ln(1.125)
f(1.25) ≈ ln(1.25)
f(1.375) ≈ ln(1.375)
f(1.5) ≈ ln(1.5)
f(1.625) ≈ ln(1.625)
f(1.75) ≈ ln(1.75)
f(1.875) ≈ ln(1.875)
f(2) ≈ ln(2)
Step 3: Calculate the approximations using the formulas for the trapezoidal and Simpson's rule.
For the trapezoidal rule:
Approximation = (h/2) * [f(a) + 2*f(x₁) + 2*f(x₂) + 2*f(x₃) + f(b)]
where a = 1, b = 2, x₁ = 1.25, x₂ = 1.5, x₃ = 1.75.
Approximation = (0.25/2) * [ln(1) + 2*ln(1.25) + 2*ln(1.5) + 2*ln(1.75) + ln(2)]
For Simpson's rule:
Approximation = (h/3) * [f(a) + 4*f(x₁) + 2*f(x₂) + 4*f(x₃) + 2*f(x₄) + 4*f(x₅) + f(b)]
where a = 1, b = 2, x₁ = 1.125, x₂ = 1.25, x₃ = 1.375, x₄ = 1.5, x₅ = 1.625.
Approximation = (0.25/3) * [ln(1) + 4*ln(1.125) + 2*ln(1.25) + 4*ln(1.375) + 2*ln(1.5) + 4*ln(1.625) + ln(1.75) + 4*ln(1.875) + ln(2)]
Step 4: Calculate the exact value of the integral.
The exact value of the integral ∫2,1 ln x dx can be evaluated using the fundamental theorem of calculus and the antiderivative of ln(x).
The antiderivative of ln(x) is x * (ln(x) - 1).
Evaluate the antiderivative at the upper limit (2) and lower limit (1) of integration:
Exact value = [2 * (ln(2) - 1)] - [1 * (ln(1) - 1)]
Step 5: Compare the approximations with the exact value of the integral.
Substitute the values in the formulas and evaluate the approximations using a calculator or software tool.
Compare the approximations obtained from the trapezoidal and Simpson's rule with the exact value obtained in Step 4.