Chapter 11: Q. 37 (page 890)

Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.
$x = a \cos t, y = b \sin t$

Short Answer

Expert verified

The curvature is $\frac{a b}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t)}^{\frac{3}{2}}} .$

Step by step solution

Step 1. Given information.

The given equations are,

$x = a \cos t, y = b \sin t$

Step 2. Curvature.

Curvature is given by,

$k = \frac{||T^{'} (t)||}{||r^{'} (t)||} ||r^{'} (t)|| = \sqrt{a^{2} \sin^{2} t + b^{2} \cos^{2} t} T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} T (t) = \frac{⟨ - a \sin t, b \cos t ⟩}{\sqrt{a^{2} \sin^{2} t + b^{2} \cos^{2} t}} T^{'} (t) = <\frac{- a b^{2} \cos t}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t)}^{\frac{3}{2}}}, \frac{a^{2} b \sin t}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t)}^{\frac{3}{2}}}> ||T^{'} (t)|| = \sqrt{\frac{a^{2} b^{4} \cos^{2} t}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t)}^{3}} + \frac{a^{4} b^{2} \sin^{2} t}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t)}^{3}}} = \sqrt{\frac{a^{2} b^{2} (b^{2} \cos^{2} t + a^{2} \sin^{2} t)}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t)}^{3}}} = \frac{a b}{a^{2} \sin^{2} t + b^{2} \cos^{2} t} k = \frac{||T^{'} (t)||}{||r^{'} (t)||} = \frac{(\frac{a b}{a^{2} \sin^{2} t + b^{2} \cos^{2} t})}{\sqrt{a^{2} \sin^{2} t + b^{2} \cos^{2} t}} = \frac{a b}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t)}^{\frac{3}{2}}}$

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Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.
$x = a \cos t, y = b \sin t$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Curvature.

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Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.x=acost,y=bsint

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Curvature.

One App. One Place for Learning.

Most popular questions from this chapter

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Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.
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