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Chapter 9: Q 10. (page 775)

Find the arc lengths of the curves defined by the parametric equations on the specified intervals.
$x = 3 t + 1$ , $y = - 2 t + 3$ , $t 鈭� [0, 1]$

Short Answer

Expert verified

The arc length is $\sqrt{13}$ units.

Step by step solution

01

Step 1. Given information.

The given parametric equation is $x = 3 t + 1$ , $y = - 2 t + 3$ .

02

Step 2. Find the derivative of the parametric equations.

$\begin{array}{rcl} x & = & 3 t + 1 \\ f' (t) & = & 3 \end{array}$ and $\begin{array}{rcl} y & = & - 2 t + 3 \\ g' (t) & = & - 2 \end{array}$

03

Step 3. Substitute the value of f't and g'(t) in arc length formula.

The arc length formula is:

${鈭�}_{a}^{b} \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} d t$ , where $[a, b] = [0, 1]$ .

${鈭�}_{a}^{b} \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} d t = {鈭�}_{0}^{1} \sqrt{{(3)}^{2} + {(- 2)}^{2}} d t$

04

Step 4. Find the arc length.

The required length of the curve is:

$\begin{array}{rcl} {鈭�}_{0}^{1} \sqrt{{(3)}^{2} + {(- 2)}^{2}} d t & = & {鈭�}_{0}^{1} \sqrt{13} d t \\ = & \sqrt{13} {鈭�}_{0}^{1} d t \\ = & \sqrt{13} {[t]}_{0}^{1} \\ = & \sqrt{13} [1 - 0] \\ = & \sqrt{13} \end{array}$

05

Step 5. Simplified answer.

Hence, the required arc length of the curve is $\sqrt{13}$ units.

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Most popular questions from this chapter

Each of the integral in exercise 38-44 represents the area of a region in a plane use polar coordinates to sketch the region and evaluate the expression

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$r = \frac{2}{1 + 3 \cos 胃}$

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