Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)1/202 sin(x2) dx, n = 4(a) the Trapezoidal Rule(b) the Midpoint Rule(c) Simpson's Rule
Question
Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)1/202 sin(x2) dx, n = 4(a) the Trapezoidal Rule(b) the Midpoint Rule(c) Simpson's Rule
Solution
The problem is to approximate the integral of 1/202 sin(x^2) dx using the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule with n = 4.
(a) The Trapezoidal Rule: The Trapezoidal Rule is given by the formula: T_n = Δx/2 * [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)] where Δx = (b - a)/n and x_i = a + iΔx.
(b) The Midpoint Rule: The Midpoint Rule is given by the formula: M_n = Δx * [f(x_1*) + f(x_2*) + ... + f(x_n*)] where Δx = (b - a)/n and x_i* = a + (i - 1/2)Δx.
(c) Simpson's Rule: Simpson's Rule is given by the formula: S_n = Δx/3 * [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)] where Δx = (b - a)/n and x_i = a + iΔx.
Note: The integral limits (a and b) are not specified in the problem. You need to know these limits to calculate the integral. Also, the function 1/202 sin(x^2) is not defined for all x, so you need to make sure that the limits of integration and the values of x used in the approximation methods are within the domain of the function.
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