Respuesta :
To use the trapezoidal rule to approximate the value of the definite integral, we need to divide the interval of integration.
In this case, from 1 to 2 we divide into n subintervals of equal length. Let the length of each subinterval be h = (2-1)/n. Then the approximate value of the definite integral using the trapezoidal rule is given by the formula:
Approximate value = (h/2) * (f(x0) + 2f(x1) + 2f(x2) + ... + 2*f(xn-1) + f(xn))
where x0, x1, x2, ..., xn are the n+1 equally spaced points in the interval from 1 to 2, and f(x) is the function being integrated (in this case, f(x) = x^2).
To use Simpson's rule to approximate the value of the definite integral, we need to divide the interval of integration into 2n subintervals of equal length (so each subinterval has length h/2). Then the approximate value of the definite integral using Simpson's rule is given by the formula:
Approximate value = (h/3) * (f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 4f(x2n-1) + f(x2n))
where x0, x1, x2, ..., x2n are the 2n+1 equally spaced points in the interval from 1 to 2, and f(x) is the function being integrated.
To compare the results of the trapezoidal and Simpson's rule approximations with the exact value of the definite integral, we need to know the exact value of the definite integral. The definite integral of x^2 from 1 to 2 is given by:
Exact value = (2³ - 1³)/3 = (8 - 1)/3 = 7/3 = 2.3333
So, to compare the results of the trapezoidal and Simpson's rule approximations with the exact value of the definite integral, we need to compute the approximate values using the formulas above for different values of n and compare them with the exact value of 2.3333.
For example, if we set n = 4 in the formulas for the trapezoidal and Simpson's rule approximations, we get:
Trapezoidal rule: Approximate value = (h/2) * (f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)) = (0.25/2) * (1² + 21.25² + 21.5² + 21.75² + 2²) = 1.9074
Simpson's rule: Approximate value = (h/3) * (f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + 4f(x5) + f(x6)) = (0.25/3) * (1² + 41.125² + 21.25² + 41.375² + 21.5² + 41.625² + 2²) = 2.
Learn more about Simpson's rule at:
brainly.com/question/28518488
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