Use the trapezoidal rule with n = 8 to approximate
∫4,1 cos(x)/x dx
Wow, this is a hard question
first of all, in the domain from 1 to 4, the graph crosses the x-axis at π/2
so we will not be able to cut it into equal trapezoids all on one side of the x-axis.
secondly, if we wanted to actually integrate cosx/x, we will have great difficulty, integration by parts doesn't work, trig substituion doesn't work, no regular methods will work.
we will have to expand cosx = 1 - x^2/2! + x^4/4! - x^6/6! + ... then divide each term by x. We then get a sequence which can be integrated and then
differentiated ending up with an infinite series.
The bot's answer is of course incorrect
with 8 intervals, we have to evaluate f(x) at nine points: 1+k/3 for k=0..8
so the value will be
1/6 (f(1) + 2f(4/3) + ... + 2f(11/3) + f(4))
so now just plug and chug.
Or, use any of various handy trapezoidal rule calculators online.
To use the trapezoidal rule to approximate the integral of a function, follow these steps:
Step 1: Determine the interval of integration. In this case, we need to integrate the function cos(x)/x over the interval from 4 to 1.
Step 2: Divide the interval into equal subintervals. In this case, n = 8, so we need to divide the interval from 4 to 1 into 8 equal subintervals. Each subinterval will have a width of (b - a) / n, where a is the lower limit (4) and b is the upper limit (1).
Width of each subinterval = (1 - 4) / 8 = -3/8
Step 3: Determine the sample points within each subinterval. We will use the left endpoints and right endpoints of each subinterval as the sample points.
Sample points for the first subinterval: x0 = 4, x1 = 4 - 3/8 = 31/8
Sample points for the second subinterval: x1 = 31/8, x2 = 31/8 - 3/8 = 29/8
And so on, until the last subinterval: xn-1 = 23/8, xn = 23/8 - 3/8 = 21/8
Step 4: Evaluate the function at each sample point and calculate the sum of the areas of the trapezoids.
Approximation = [(b - a) / (2 * n)] * [f(a) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(b)]
In this case, f(x) = cos(x)/x
Approximation = [-3/8 * (cos(4)/4) + 2(cos(31/8)/(31/8)) + 2(cos(29/8)/(29/8)) + ... + 2(cos(23/8)/(23/8)) + (cos(21/8)/(21/8))]
You can simplify this expression further by plugging in the values of the sample points and evaluating the trigonometric functions. Finally, calculate the sum and multiply it with the width of each subinterval.