Calculate the average over the given interval. f(x)=sin(pi/x)/x^2, [1,2]
Let u = 1/x
du = -dx/x^2
Int(sin(pi/x)/x^2 dx)[1,2]
= Int(-sin(pi*u) du)[1,.5]
= 1/pi cos(pi*u) [1,.5]
= 1/pi(cos pi/2 - cos pi)
= 1/pi(0+1)
= 1/pi
Since the interval length is 1, the average is just 1/pi
To calculate the average value of a function over an interval, we need to find the definite integral of the function over that interval and then divide by the width of the interval.
The definite integral of the function f(x) = sin(pi/x)/x^2 over the interval [1,2] can be written as:
∫[1,2] (sin(pi/x)/x^2) dx.
To solve this integral, it requires integration techniques such as substitution or integration by parts. However, these techniques can be quite complex when dealing with the given function.
Alternatively, we can approximate the average value using numerical methods, such as the midpoint rule, trapezoidal rule, or Simpson's rule. These methods divide the interval into smaller subintervals and evaluate the function at different points within each subinterval to estimate the integral. Let's use the trapezoidal rule in this example.
For simplicity, we'll divide the interval [1,2] into 4 subintervals: [1, 1.25, 1.5, 1.75, 2]. The width of each subinterval will be (2 - 1) / 4 = 0.25.
Now we evaluate the function at each endpoint and the midpoint of each subinterval:
f(1) = sin(pi/1)/(1^2) = sin(pi)
f(1.25) = sin(pi/1.25)/(1.25^2)
f(1.5) = sin(pi/1.5)/(1.5^2)
f(1.75) = sin(pi/1.75)/(1.75^2)
f(2) = sin(pi/2)/(2^2) = sin(pi)/4
Next, we use the trapezoidal rule formula:
Average ≈ (f(1) + 2*f(1.25) + 2*f(1.5) + 2*f(1.75) + f(2))/6
Plugging in the values we evaluated:
Average ≈ (sin(pi) + 2*f(1.25) + 2*f(1.5) + 2*f(1.75) + sin(pi)/4)/6
Now, we compute f(1.25), f(1.5), and f(1.75):
f(1.25) = sin(pi/1.25)/(1.25^2)
f(1.5) = sin(pi/1.5)/(1.5^2)
f(1.75) = sin(pi/1.75)/(1.75^2)
After substituting these values, we can simplify the expression and calculate the average value of the function over the interval [1,2].
To calculate the average of a function over a given interval, we need to find the definite integral of the function over that interval and then divide by the length of the interval.
In this case, we are given the function f(x) = sin(pi/x)/x^2, and we need to find the average of this function over the interval [1, 2].
Step 1: Find the definite integral of the function over the interval [1, 2].
To find the integral of f(x) with respect to x, we can use various integration techniques such as substitution or integration by parts. However, in this case, the integral of f(x) is not an elementary function, and it is difficult to find an exact value. So, we need to use numerical methods to approximate the integral.
One common numerical method is the trapezoidal rule. This rule approximates the area under a curve by dividing the interval into small trapezoids and summing up their areas.
To use the trapezoidal rule, we first need to divide the interval [1, 2] into smaller subintervals. Let's say we divide it into n subintervals. Then, the width of each subinterval, Δx, is given by Δx = (2-1)/n = 1/n.
Next, we evaluate the function at the endpoints of each subinterval and multiply them by the corresponding widths. Let's denote the function evaluated at the left endpoint of the i-th subinterval as f(x_i), where x_i = 1 + (i-1)Δx. Then, the area of each trapezoid is given by (f(x_i-1)+f(x_i))/2 * Δx.
Finally, we sum up the areas of all the trapezoids to get an approximation of the integral. The formula for the approximation is:
Approximation of the integral = Δx/2 * (f(x_0) + 2*f(x_1) + 2*f(x_2) + ... + 2*f(x_{n-1}) + f(x_n))
For a more accurate approximation, we can increase the value of n.
Step 2: Divide the approximation of the integral by the length of the interval.
Now that we have an approximation of the integral, we divide it by the length of the interval [1, 2] to get the average.
Length of the interval = 2 - 1 = 1
Average = Approximation of the integral / Length of the interval
By following these steps, you can calculate the average of the function f(x) = sin(pi/x)/x^2 over the interval [1, 2].