Question

In Exercises \(27-29,\) find the length of the nonsmooth curve. $$y=x^{3}+5|x| \quad \text { from } x=-2 \text { to } x=1$$

Short answer

The length of the curve \(y=x^{3}+5|x|\) from \(x=-2\) to \(x=1\) is obtained by adding the lengths of the curve over the two intervals \([-2,0]\) and \([0,1]\). Each length is derived from the formula: \(\int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx\), where \(f'(x)\) is the derivative of the function in that interval.

Step-by-step solution

Identify Functions & Derivatives for Different Intervals

For the interval \(-2 \leq x \leq 0\), the function becomes \(y = x^3 - 5x\), as \(|x|\) equals \(-x\) when \(x\) is less than 0. The derivative is \(dy/dx = 3x^2 - 5\).\n For the interval \(0 \leq x \leq 1\), the function becomes \(y = x^3 + 5x\), as \(|x|\) equals \(x\) when \(x\) is greater than or equal to 0. The derivative here is \(dy/dx = 3x^2 + 5\).

Compute Length for Each Interval

The length of the curve for each interval is calculated using the formula \(\int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx\).\n For x in \([-2,0]\), the length \(L1\) is given by: \(\int_{-2}^{0} \sqrt{1 + (3x^2 - 5)^2} dx\).\n For x in \([0,1]\), the length \(L2\) is given by: \(\int_{0}^{1} \sqrt{1 + (3x^2 + 5)^2} dx\).

Total Length of Curve

The total length of the curve is obtained by summing the lengths calculated for the two intervals, that is, \(L = L1 + L2\).\nThis requires careful calculation, as it may involve complex integration.