What is the length of the curve \( y=5 x^{3 / 2} \) from \( x=0 \) to \( x=4 \) ? Express your answer to 3 (2 marks)

To find the length of the curve given by the equation \( y = 5x^{3/2} \) from \( x = 0 \) to \( x = 4 \), we can use the formula for the arc length of a curve defined by a function \( y = f(x) \): \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] First, we need to compute the derivative \( \frac{dy}{dx} \): \[ y = 5x^{3/2} \] Using the power rule, we find: \[ \frac{dy}{dx} = 5 \cdot \frac{3}{2} x^{1/2} = \frac{15}{2} x^{1/2} \] Next, we calculate \( \left( \frac{dy}{dx} \right)^2 \): \[ \left( \frac{dy}{dx} \right)^2 = \left( \frac{15}{2} x^{1/2} \right)^2 = \frac{225}{4} x \] Now, we can substitute this into the arc length formula: \[ L = \int_0^4 \sqrt{1 + \frac{225}{4} x} \, dx \] Next, we simplify the expression under the square root: \[ L = \int_0^4 \sqrt{1 + \frac{225}{4} x} \, dx \] To evaluate this integral, we can use a substitution. Let: \[ u = 1 + \frac{225}{4} x \quad \Rightarrow \quad du = \frac{225}{4} \, dx \quad \Rightarrow \quad dx = \frac{4}{225} \, du \] Now we need to change the limits of integration. When \( x = 0 \): \[ u = 1 + \frac{225}{4} \cdot 0 = 1 \] When \( x = 4 \): \[ u = 1 + \frac{225}{4} \cdot 4 = 1 + 225 = 226 \] Now we can rewrite the integral in terms of \( u \): \[ L = \int_1^{226} \sqrt{u} \cdot \frac{4}{225} \, du \] This simplifies to: \[ L = \frac{4}{225} \int_1^{226} u^{1/2} \, du \] Now we compute the integral: \[ \int u^{1/2} \, du = \frac{u^{3/2}}{\frac{3}{2}} = \frac{2}{3} u^{3/2} \] Evaluating this from 1 to 226: \[ \left[ \frac{2}{3} u^{3/2} \right]_1^{226} = \frac{2}{3} \left( 226^{3/2} - 1^{3/2} \right) \] Calculating \( 226^{3/2} \): \[ 226^{3/2} = 226 \sqrt{226} \] Now we can approximate \( \sqrt{226} \): \[ \sqrt{226} \approx 15.033 \] Thus, \[ 226^{3/2} \approx 226 \cdot 15.033 \approx 3392.478 \] Now substituting back into the integral: \[ \frac{2}{3} \left( 3392.478 - 1 \right) = \frac{2}{3} \cdot 3391.478 \approx 2260.985 \] Finally, we multiply by \( \frac{4}{225} \): \[ L \approx \frac{4}{225} \cdot 2260.985 \approx 40.000 \] Thus, the length of the curve from \( x = 0 \) to \( x = 4 \) is approximately: \[ \boxed{40.000} \]