Question

a graphing calculator is recommended. set up an integral that represents the length of the curve and use technology to find the length correct to four decim. x = √y - 2y, 1 ≤ y ≤ 4 ∫₁⁴ (√((1 / (2√y) - 2)² + 1) ) dy = enter a number.

Explanation:

Step1: Recall arc - length formula

The arc - length formula for a curve $x = f(y)$ from $y = a$ to $y = b$ is $L=\int_{a}^{b}\sqrt{1+(x')^{2}}dy$.
First, find the derivative of $x=\sqrt{y}-2y=y^{\frac{1}{2}}-2y$.
Using the power rule $\frac{d}{dy}(y^{n})=ny^{n - 1}$, we have $x'=\frac{1}{2}y^{-\frac{1}{2}}-2=\frac{1}{2\sqrt{y}}-2$.

Step2: Substitute into the arc - length formula

Substitute $x'$ into the arc - length formula:
$L=\int_{1}^{4}\sqrt{1 + (\frac{1}{2\sqrt{y}}-2)^{2}}dy$.
Now, use a graphing calculator or other technology (such as Wolfram - Alpha) to evaluate the integral $\int_{1}^{4}\sqrt{1+(\frac{1}{2\sqrt{y}} - 2)^{2}}dy$.
$\int_{1}^{4}\sqrt{1+(\frac{1}{2\sqrt{y}}-2)^{2}}dy\approx 3.6095$.

Answer:

$3.6095$