Find the half range sine for the function 𝑓(𝑥) = 𝑥 in 0 ≤ 𝑥 ≤ 𝑙

To find the half range sine for the function 𝑓(𝑥) = 𝑥 in 0 ≤ 𝑥 ≤ 𝑙, we need to follow these steps:

1. Determine the period of the function: Since 𝑓(𝑥) = 𝑥 is a linear function, it does not have a periodic behavior. Therefore, the period is infinite.

2. Find the half range: In this case, the given range is 0 ≤ 𝑥 ≤ 𝑙. To find the half range, we divide the given range by 2. So, the half range is 0 ≤ 𝑥 ≤ 𝑙/2.

3. Calculate the integral: To find the half range sine, we need to calculate the integral of the function 𝑓(𝑥) = 𝑥 over the half range 0 ≤ 𝑥 ≤ 𝑙/2. The integral of 𝑥 with respect to 𝑥 is (1/2)𝑥^2. So, the integral of 𝑓(𝑥) = 𝑥 over the half range is (1/2)(𝑥^2) evaluated from 0 to 𝑙/2.

4. Evaluate the integral: Plugging in the upper limit 𝑙/2 into the integral expression, we get (1/2)((𝑙/2)^2) = (1/2)(𝑙^2/4) = 𝑙^2/8. Plugging in the lower limit 0 into the integral expression, we get (1/2)(0^2) = 0. Therefore, the value of the integral over the half range is 𝑙^2/8 - 0 = 𝑙^2/8.

5. Simplify the result: The value of the integral over the half range is 𝑙^2/8.

So, the half range sine for the function 𝑓(𝑥) = 𝑥 in 0 ≤ 𝑥 ≤ 𝑙 is 𝑙^2/8.