difinite integral f(x)=1/y^2dy from x=1 to x=4
To evaluate the definite integral of f(x) = 1/y^2 dy from x = 1 to x = 4, you need to follow these steps:
1. Determine the limits of integration: x = 1 and x = 4.
2. Rewrite the integral using the appropriate variable substitution. In this case, we substitute y for x, since we are integrating with respect to y. The new limits of integration become y = 1 and y = 4.
3. Evaluate the integral. The integral of 1/y^2 dy is -1/y.
Now let's go through the steps:
1. The limits of integration are x = 1 and x = 4.
2. Since we are integrating with respect to y, we need to express the limits in terms of y. To do this, we substitute y for x:
When x = 1, y = 1.
When x = 4, y = 4.
So the new limits of integration are y = 1 and y = 4.
3. Evaluate the integral: ∫ (from 1 to 4) 1/y^2 dy.
The integral of 1/y^2 dy is -1/y.
Now we can substitute the limits of integration:
∫ (from 1 to 4) 1/y^2 dy = [-1/y] (from 1 to 4)
= [-1/4] - [-1/1]
= -1/4 + 1
= 3/4.
Therefore, the value of the definite integral ∫ (from 1 to 4) 1/y^2 dy is 3/4.