Chapter 9: Q 10. (page 775)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

Step 2. Find the derivative of the parametric equations.

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

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

The arc length formula is:

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

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

Step 4. Find the arc length.

The required length of the curve is:

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

Step 5. Simplified answer.

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the parametric equations.

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

Step 4. Find the arc length.

Step 5. Simplified answer.

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the parametric equations.

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

Step 4. Find the arc length.

Step 5. Simplified answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

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