Find the length of the curve y^2=x^3 from the origin where the tangent line makes an angle of 45 degrees with the x-axis.
since y=x^(3/2)
y' = (3/2)x^(1/2)
y'=1 when x = 4/9 and y = 8/27
so, the arc length is
s = ∫[0,4/9) √(1+((3/2)x^(1/2))^2) dx
= 1/2 ∫[0,4/9) √(9x+4) dx
= 8/27 (2√2-1)
As always, check my math
To find the length of the curve y^2=x^3 from the origin where the tangent line makes an angle of 45 degrees with the x-axis, we can use calculus and parametric equations.
First, let's parameterize the curve. We can let x = t^2 and y = t^3/2, where t is a parameter.
Next, we can find the derivative dy/dx by taking the derivative of y with respect to t and dividing it by the derivative of x with respect to t:
dy/dx = (dy/dt) / (dx/dt)
Taking the derivatives, we get:
dy/dt = (3/2)t^(3/2 - 1) = (3/2)t^(1/2)
dx/dt = 2t
Therefore, dy/dx = (3/2)t^(1/2) / 2t = (3/4)t^(-1/2)
We want to find the value of t such that the tangent line has an angle of 45 degrees with the x-axis. We know that the slope of a line with an angle of 45 degrees is 1. So, we set dy/dx equal to 1:
(3/4)t^(-1/2) = 1
(3/4) = t^(-1/2)
Now, we can solve for t:
t^(-1/2) = 4/3
(1/t)^(1/2) = 4/3
1/t = (4/3)^2
1/t = 16/9
t = 9/16
Now, we can substitute this value of t back into the parametric equations:
x = (9/16)^2 = 81/256
y = (9/16)^(3/2) = 27/64
So, the point on the curve where the tangent line makes an angle of 45 degrees with the x-axis is (81/256, 27/64).
To find the length of the curve from the origin to this point, we use the arc length formula:
L = ∫(a to b) sqrt(1 + (dy/dx)^2) dx
In this case, a is 0 (the origin) and b is 81/256. So, we have:
L = ∫(0 to 81/256) sqrt(1 + [(3/4)t^(-1/2)]^2) dx
We substitute dx = 2t dt (from the parametric equations), and simplify the integrand:
L = ∫(0 to 81/256) sqrt(1 + (9/16t)^2) [2t dt]
L = 2∫(0 to 81/256) sqrt(1 + 81/256t^2) dt
At this point, we can simplify the integrand, but the integration itself is not straightforward. It requires specialized techniques such as elliptic integrals. So, obtaining the exact value of the length of the curve would be a complex and lengthy process.
To approximate the length numerically, we can use numerical integration methods like Simpson's rule or the trapezoidal rule. These methods can be applied to find an approximation of the integral and provide an approximate value for the length of the curve.