How does 114 become 4 from ∫[0,114](√1+4x^2/57^2)dx become ∫[0,4](√1+u^2(57/2du))
If u=2x/57 then
du = 2/57 dx ---> dx = 57/2 du
If x=114 then u=4
To understand how the integral expression changes from the first equation to the second equation, let's analyze the steps involved:
1. Original Integral Expression:
∫[0, 114] (√(1 + 4x^2/57^2)) dx
2. Variable Substitution:
Let's introduce a new variable "u" to simplify the expression. We can make the substitution:
u = (57/2) * x
To solve for "x" in terms of "u," rearrange the equation:
x = (2/57) * u
This substitution is known as "u-substitution" and helps in simplifying integrals by changing the variable of integration.
3. Change the Limits of Integration:
When we substitute the variable "u," we need to change the limits of integration as well. So, plug in the original limits of integration (0 and 114) into the new equation to find the new limits.
For the lower limit:
x = (2/57) * u
x = (2/57) * 0 = 0
For the upper limit:
x = (2/57) * u
x = (2/57) * 114 = 4
Therefore, the new limits of integration become [0, 4].
4. Integral in terms of "u":
Now, substitute the new variable "u" and its corresponding limits in the original integral expression.
∫[0, 114] (√(1 + 4x^2/57^2)) dx
becomes
∫[0, 4] (√(1 + u^2)) (57/2) du
The resulting integral expression now contains the new variable "u" and the modified limits of integration [0, 4].
In summary, by introducing the variable substitution u as (57/2) * x, and updating the limits of integration accordingly, the original integral expression ∫[0, 114] (√(1 + 4x^2/57^2)) dx transforms into the new integral expression ∫[0, 4] (√(1 + u^2)) (57/2) du.