Find the length of the parabola y^2=4ax cut off the line x=a about the x-axis
do you want the arc length (above and below the x-axis), or the surface when the arc (above or below the axis) is rotated about the x-axis?
2^1/2/a^1/2log|a+1/2|
√2/√a+log|a+√2|
To find the length of the parabola y^2 = 4ax cut off by the line x = a about the x-axis, we need to calculate the arc length of the curve.
The formula to calculate the length of a curve is given by the arc length formula:
L = ∫[a, b] √[1 + (dy/dx)^2] dx,
where a and b are the x-coordinates of the endpoints of the curve.
First, let's find the equation of the curve that represents the parabola y^2 = 4ax. Solving for y, we get:
y = ± √(4ax).
Since we are interested in the part of the curve cut off by the line x = a about the x-axis, we only need to consider the positive value of y.
Now, let's find dy/dx. Differentiating y = √(4ax) with respect to x, we get:
dy/dx = (2a)/(√(4ax)).
Substituting this back into the arc length formula, we have:
L = ∫[a, b] √[1 + ((2a)/(√(4ax)))^2] dx.
To integrate the above expression, we can simplify it first. Expanding the square inside the square root, we get:
L = ∫[a, b] √[1 + 4a^2/(4ax)] dx
= ∫[a, b] √[1 + a/(ax)] dx
= ∫[a, b] √[1 + 1/x] dx.
Next, we can rationalize the square root expression by multiplying the numerator and the denominator by √x:
L = ∫[a, b] √[(x + 1)/x] dx.
Now, we can simplify the expression further by pulling out the square root of x from the square root denominator:
L = ∫[a, b] (√(x + 1)/√x) dx.
Finally, we can simplify the expression even more:
L = ∫[a, b] (√(1 + 1/x)) dx.
To evaluate this integral, we need to know the value of a and the bounds a and b, which were not specified in the question. Once we have those values, we can solve the integral using standard integration techniques, such as substitution or integration by parts.
So, to find the length of the parabola y^2 = 4ax cut off by the line x = a about the x-axis, follow the steps explained above and use the appropriate values of a and b in the integral to calculate the length.