Problem 54 - Essential Calculus

Question

Find the arc length of the curve over the given interval. $$ y=\frac{1}{2} x^{2}, \quad[0,4] $$

Step-by-Step Solution

Answer

The arc length of the curve \(y=\frac{1}{2} x^{2}\) over the interval \([0,4]\) is approximately 8 units.

Step 1: Derive the Function

The derivative of the function \(f(x) = \frac{1}{2} x^2\) is \(f'(x) = \frac{d}{dx} (\frac{1}{2} x^2) = x\).

Step 2: Use the Arc Length Formula

Plug \(f'(x) = x\) into the arc length formula. This gives \(L= \int_{a}^{b} \sqrt{1+(x)^{2}} dx\). In this case, the interval \([a,b]\) is \([0,4]\). So, we need to evaluate the integral from 0 to 4.

Step 3: Evaluate the Integral

To evaluate the integral, we can use a table of integrals or a calculus calculator. Doing so, we find \(\int_{0}^{4} \sqrt{1+(x)^{2}} dx \approx 8.37\). So, the arc length of the curve is approximately 8.37 units.

Step 4: Round the Answer

Based on the significant figures rule, we should round the arc length to the nearest whole number. So, the final answer for the arc length of the curve over the interval \([0,4]\) is approximately 8 units.