Problem 61 - Essential Calculus

Question

Find the length of the curve \(y^{2}=x^{3}\) from the origin to the point where the tangent makes an angle of \(45^{\circ}\) with the \(x\) -axis.

Step-by-Step Solution

Answer

The length of the curve from the origin to the given point can be found by performing the integration in Step 3. As the integration is somewhat complex, it might need the use of software or a table of integrals. The solution is the definite integral of \(\sqrt{1+\left(\frac{dy}{dx}\right)^2} dx\) from \(0\) to \(\frac{2^{1/3}}{3^{2/3}}\).

Step 1: Derive the given function and Formula for Curve Length

Derive the function \(y^{2}=x^{3}\) to get \(2y \frac{dy}{dx}=3x^{2}\). Hence, \(\frac{dy}{dx} =\frac{3x^{2}}{2y}\). The formula for the length \(s\) of a curve given by \(y=f(x)\) from \(x=a\) to \(x=b\) is \(s=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx\).

Step 2: Find Tangent Angle

Knowing that the tangent of the angle \(θ\) between the tangent to the curve and the x-axis is given by the derivative of the function \(tan(\theta) = \frac{dy}{dx}\), set \(tan(\theta) = \frac{3x^{2}}{2y} = 1\) (since for \(45^{\circ}\), \(tan(\theta) = 1)\). Solving for \(x\), we get \(x= \frac{2^{1/3}}{3^{2/3}}\).

Step 3: Find Curve Length

Substitute this \(x\) -value along with \(a = 0\) and \(b = \frac{2^{1/3}}{3^{2/3}}\) and \( \frac{dy}{dx} =\frac{3x^{2}}{2y}\) into the formula \(s=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx\) and carry out the integration to obtain the length.