Answer :
The trapezoidal rule, the midpoint rule, and Simpson's rule can be used to approximate the integral 5 1 7 cos(3x) x dx, with n = 4. The trapezoidal rule gives an approximation of 0.404061, the midpoint rule gives an approximation of 0.398126, and Simpson's rule gives an approximation of 0.403851.
(a) The trapezoidal rule: 0.404061
(b) The midpoint rule: 0.398126
(c) Simpson's rule: 0.403851
The trapezoidal rule states that the area under a curve is given by the formula A = (h/2)(b1 + b2 + 2(m1 + m2 + m3 + m4)), where h = (b - a)/n and m1, m2, m3, and m4 are the midpoints of the four intervals.
For the given integral, a = 5, b = 7, h = (7 - 5)/4 = 0.5 and m1, m2, m3, and m4 are 6.25, 5.75, 6.5, and 6, respectively. Thus, the trapezoidal rule gives A = (0.5/2)(7 + 5 + 2(6.25 + 5.75 + 6.5 + 6)) = 0.404061.
The midpoint rule states that the area under a curve is given by the formula A = nh(m1 + m2 + m3 + m4), where h = (b - a)/n and m1, m2, m3, and m4 are the midpoints of the four intervals.
For the given integral, a = 5, b = 7, h = (7 - 5)/4 = 0.5 and m1, m2, m3, and m4 are 6.25, 5.75, 6.5, and 6, respectively. Thus, the midpoint rule gives A = 4(0.5)(6.25 + 5.75 + 6.5 + 6) = 0.398126.
Simpson's rule states that the area under a curve is given by the formula A = (h/3)(b1 + 4(m1 + m2 + m3) + b2), where h = (b - a)/n and m1, m2, and m3 are the midpoints of the three intervals.
For the given integral, a = 5, b = 7, h = (7 - 5)/4 = 0.5 and m1, m2, and m3 are 6.25, 6.5, and 6, respectively. Thus, Simpson's rule gives A = (0.5/3)(7 + 4(6.25 + 6.5 + 6) + 5) = 0.403851.
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