Use Simpson's rule with n = 4 to approximate. Keep at least 2 decimal places accuracy.
Integrate: (cos(x))/(x) x=1 to 5
To use Simpson's rule with n = 4 to approximate the integral of (cos(x))/(x) from x = 1 to 5, follow these steps:
1. Calculate h, the step size:
h = (b - a) / n
where a is the lower limit (1) and b is the upper limit (5), and n is the number of intervals (4).
h = (5 - 1) / 4
h = 1
2. Calculate the function values at the points of integration:
Let x0 = 1, x1 = 2, x2 = 3, x3 = 4, and x4 = 5.
f(x0) = (cos(1))/(1)
f(x1) = (cos(2))/(2)
f(x2) = (cos(3))/(3)
f(x3) = (cos(4))/(4)
f(x4) = (cos(5))/(5)
Evaluating these values using a calculator, we get:
f(x0) ≈ 0.540302
f(x1) ≈ -0.416147
f(x2) ≈ -0.145500
f(x3) ≈ -0.653644
f(x4) ≈ 0.283662
3. Apply Simpson's rule formula:
Approximation = (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + f(x4)]
Approximation ≈ (1/3) * [0.540302 + 4(-0.416147) + 2(-0.145500) + 4(-0.653644) + 0.283662]
Approximation ≈ (1/3) * [0.540302 - 1.664588 - 0.291000 - 2.614576 + 0.283662]
Approximation ≈ (1/3) * [-3.746200]
Approximation ≈ -1.248733
Therefore, using Simpson's rule with n = 4, the approximate value of the integral (cos(x))/(x) from x = 1 to 5 is -1.25 (rounded to two decimal places).
To solve this problem using Simpson's rule, we need to follow these steps:
1. Divide the interval [1, 5] into n equal subintervals. In this case, n = 4, so we divide the interval into 4 subintervals: [1, 2], [2, 3], [3, 4], and [4, 5].
2. Calculate the step size, h, which is the width of each subinterval. In this case, h = (5-1)/4 = 1.
3. Evaluate the function f(x) = (cos(x))/(x) at the endpoints and midpoint of each subinterval. In our case, we need to evaluate f(1), f(2), f(3), f(4), and f(5).
- f(1) = (cos(1))/(1) = 0.5403
- f(2) = (cos(2))/(2) = 0.1288
- f(3) = (cos(3))/(3) = -0.0859
- f(4) = (cos(4))/(4) = -0.0787
- f(5) = (cos(5))/(5) = 0.0807
4. Calculate the approximation using the Simpson's rule formula for integration:
- Approximation = (h/3) * [ f(1) + 4*f(2) + 2*f(3) + 4*f(4) + f(5) ]
- Approximation = (1/3) * [0.5403 + 4*0.1288 + 2*(-0.0859) + 4*(-0.0787) + 0.0807 ]
- Approximation = (1/3) * [0.5403 + 0.5152 - 0.1718 - 0.3148 + 0.0807 ]
- Approximation = (1/3) * [0.6500]
- Approximation = 0.2167 (rounded to 4 decimal places)
Therefore, using Simpson's rule with n = 4, the approximation for the integral of (cos(x))/(x) from x = 1 to 5 is 0.2167 with at least 2 decimal places accuracy.
see a nice calculator here to check your work:
http://www.emathhelp.net/calculators/calculus-2/simpsons-rule-calculator/