Chapter 14: Q. 14 (page 1096)

Write $\int_{C} F (x, y, z) \cdot d r$ explicitly as an integral of t, where $F (x, y, z) = (z y - x, x z - y, x y - z)$ and role="math" localid="1650820575074" $r (t) = (t, \cos t, \sin t)$ for $a \leq t \leq b$ .

Short Answer

Expert verified

Hence, $\int_{C} F (x, y, z) \cdot d r = \int_{a}^{b} \sin t \cos t - t - t \cos 2 t d t$

Step by step solution

Step 1. Given Information

Write $\int_{C} F (x, y, z) \cdot d r$ explicitly as an integral of t, where role="math" localid="1650822686518" $F (x, y, z) = (z y - x, x z - y, x y - z)$ and role="math" localid="1650821798839" $r (t) = (t, \cos t, \sin t)$ for $a \leq t \leq b$ .

Step 2. The curve parameterize by r(t)=(t, cost, sint) at a≤t≤b

Thus $r' (t) = (1, - \sin t, \cos t)$ . So,

$\int_{C} f (x, y, z) dr = \int_{a}^{b} f (r (t) \cdot r' (t) d t \int_{C} F (x, y, z) \cdot dr = \int_{a}^{b} (\sin t \cos t - t, t \sin t - \cos t, t \cos t - \sin t) (1, - \sin t, \cos t) d t \int_{C} F (x, y, z) \cdot d r = \int_{a}^{b} 1 (\sin t \cos t - t) - \sin t (t \sin t - \cos t) + \cos t (t \cos t - \sin t) d t \int_{C} F (x, y, z) \cdot d r = \int_{a}^{b} \sin t \cos t - t - t \sin^{2} t + \sin t \cos t + t \cos^{2} t - \sin t \cos t d t \int_{C} F (x, y, z) \cdot d r = \int_{a}^{b} \sin t \cos t - t + t (\cos^{2} t - \sin^{2} t) d t \int_{C} F (x, y, z) \cdot d r = \int_{a}^{b} \sin t \cos t - t - t \cos 2 t d t$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Write $\int_{C} F (x, y, z) \cdot d r$ explicitly as an integral of t, where $F (x, y, z) = (z y - x, x z - y, x y - z)$ and role="math" localid="1650820575074" $r (t) = (t, \cos t, \sin t)$ for $a \leq t \leq b$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The curve parameterize by r(t)=(t, cost, sint) at a≤t≤b

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Write ∫CF(x,y,z)·dr explicitly as an integral of t, where F(x,y,z)=(zy−x,xz−y,xy−z) and role="math" localid="1650820575074" r(t)=(t,cost,sint) for a≤t≤b.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The curve parameterize by r(t)=(t, cost, sint) at a≤t≤b

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Write $\int_{C} F (x, y, z) \cdot d r$ explicitly as an integral of t, where $F (x, y, z) = (z y - x, x z - y, x y - z)$ and role="math" localid="1650820575074" $r (t) = (t, \cos t, \sin t)$ for $a \leq t \leq b$ .