Which of the following integrals represents the length of the curve y equals the square root of x from x = 0 to x = 3? (15 points)
A) the integral from 0 to 3 of the square root of 1 plus 1 over 4 times x, dx
B) the integral from 0 to 3 of the square root of 1 plus 1 over 2 times x, dx
C) the integral from 0 to 3 of the square root of 1 plus x, dx
D) None of these
well, d/dx √x = 1/(2√x)
so what do you think?
To determine which of the given integrals represents the length of the curve y = √x from x = 0 to x = 3, we can apply the arc length formula.
The arc length formula for a function y = f(x) from x = a to x = b is given by:
L = ∫[a to b] √(1 + (dy/dx)^2) dx
In this case, the function y = √x, so dy/dx = 1/(2√x). Substituting this into the arc length formula, we have:
L = ∫[0 to 3] √(1 + (1/(2√x))^2) dx
Simplifying, we get:
L = ∫[0 to 3] √(1 + 1/(4x)) dx
So, the correct integral representing the length of the curve y = √x from x = 0 to x = 3 is option A) the integral from 0 to 3 of √(1 + 1/(4x)), dx.