QQuestionMathematics
QuestionMathematics
"Estimate the integral ∫301x^3 + 1−−−−−√dx
by the trapezoidal rule using n = 4."
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Answer
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Step 1:: Understand the problem and gather necessary information
We are asked to estimate the integral ∫(301)x^3 + 1 dx from 0 to 1 using the trapezoidal rule with n = 4.
Step 2:: Set up the trapezoidal rule formula
∫(f(x)) dx ≈ T_n = rac{Δx}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]
The trapezoidal rule formula is given by: where Δx = (b - a) / n, and in our case, a = 0, b = 1, and n = 4.
Step 3:: Calculate Δx
Δx = rac{(b - a)}{n} = rac{1 - 0}{4} = rac{1}{4}
Step 4:: Calculate the function values at the given points
f(x_4) = (301)(1)3+1 = 302
We need to find the function values at x^0 = 0, x^1 = Δx, x^2 = 2Δx, x^3 = 3Δx, and x^4 = 1. So,
Step 5:: Apply the trapezoidal rule formula
T_4 = rac{1/4}{2} [1 + 2(1.4645) + 2(2.8780) + 2(4.2400) + 302]
Now, we can apply the trapezoidal rule formula: Substitute the values:
Step 6:: Calculate the approximate integral value
T_4 ≈ rac{1/4}{2} (1 + 2(1.4645) + 2(2.8780) + 2(4.2400) + 302) ≈ 126.1573
Calculate the sum and multiply by Δx/ 2:
Final Answer
The approximate value of the integral ∫(301)x^3 + 1 dx from 0 to 1 using the trapezoidal rule with n = 4 is approximately 126.1573.
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