Chapter 14: Q. 50 (page 1107)

Evaluate the line integrals in Exercises 45–50.
$\int_{C} f (x, y, z) d s, where f (x, y, z) = (x + z) 3^{y}$ and C is the curve parameterized by role="math" localid="1651219517485" $x = 2 - t, y = 4 t, z = t + 5, for t = 1 to t = 4$ .

Short Answer

Expert verified

The value of line integral is $\int_{C} f (x, y, z) d s \approx 94, 500$

Step by step solution

Step 1. Given Information

We have to evaluate the line integrals in given exercise.

$\int_{C} f (x, y, z) d s, where f (x, y, z) = (x + z) 3^{y}$ andC is the curve parameterized by $x = 2 - t, y = 4 t, z = t + 5, for t = 1 to t = 4$ .

Step 2. To find the line integral ∫Cf(x,y,z)ds where r(t)=(2-t,4t,t+5)

Here we have

$f (x (t), y (t), z (t)) = (2 - t + t + 5) 3^{4 t} f (x (t), y (t), z (t)) = (2 + 5) 3^{4 t} f (x (t), y (t), z (t)) = 73^{4 t}$

and

$\frac{d s}{d t} = \frac{d}{d t} (2 - t, 4 t, t + 5) \frac{d s}{d t} = (\frac{d}{d t} (2 - t), \frac{d}{d t} 4 t, \frac{d}{d t} (t + 5)) \frac{d s}{d t} = (- 1, 4, 1)$

Step 3. Thus, ∫Cf(x,y,z)ds=7∫1434t(-1,4,1)dt

$\int_{C} f (x, y, z) d s = 7 \int_{1}^{4} 3^{4 t} (- 1, 4, 1) d t \int_{C} f (x, y, z) d s = 7 (- \int_{1}^{4} 3^{4 t} d t + 4 \int_{1}^{4} 3^{4 t} d t + \int_{1}^{4} 3^{4 t} d t) l e t 4 t = u 4 d t = d u d t = \frac{1}{4} d u \int_{C} f (x, y, z) d s = 7 (- \frac{1}{4} \int_{1}^{4} 3^{u} du + \frac{4}{4} \int_{1}^{4} 3^{u} du + \frac{1}{4} \int_{1}^{4} 3^{u} du) \int_{C} f (x, y, z) d s = 7 {(- \frac{1}{4} [\frac{3^{u}}{\log 3}] + 1 [\frac{3^{u}}{\log 3}] + \frac{1}{4} [\frac{3^{u}}{\log 3}])}_{1}^{4} \int_{C} f (x, y, z) d s = 7 {[\frac{3^{u}}{\log 3}]}_{1}^{4} \int_{C} f (x, y, z) d s = 7 {[\frac{3^{4 t}}{\log 3}]}_{1}^{4} \int_{C} f (x, y, z) d s = 7 [\frac{3^{4 \cdot 4}}{\log 3} - \frac{3^{4 \cdot 1}}{\log 3}] \int_{C} f (x, y, z) d s = 7 \cdot \frac{3^{4}}{\log 3} [3^{4} - 1] \int_{C} f (x, y, z) d s = 7 \cdot \frac{3^{4}}{\log 3} [81 - 1] \int_{C} f (x, y, z) d s = 7 \cdot \frac{81}{\log 3} \cdot 80 \int_{C} f (x, y, z) d s \approx \frac{45, 360}{0.48} \int_{C} f (x, y, z) d s \approx 94, 500$

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.

Evaluate the line integrals in Exercises 45–50.
$\int_{C} f (x, y, z) d s, where f (x, y, z) = (x + z) 3^{y}$ and C is the curve parameterized by role="math" localid="1651219517485" $x = 2 - t, y = 4 t, z = t + 5, for t = 1 to t = 4$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. To find the line integral ∫Cf(x,y,z)ds where r(t)=(2-t,4t,t+5)

Step 3. Thus, ∫Cf(x,y,z)ds=7∫1434t(-1,4,1)dt

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Geometry

Discrete Mathematics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

Evaluate the line integrals in Exercises 45–50.∫Cf(x,y,z)ds,wheref(x,y,z)=(x+z)3y and C is the curve parameterized by role="math" localid="1651219517485" x=2−t,y=4t,z=t+5,fort=1tot=4.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. To find the line integral ∫Cf(x,y,z)ds where r(t)=(2-t,4t,t+5)

Step 3. Thus, ∫Cf(x,y,z)ds=7∫1434t(-1,4,1)dt

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Geometry

Discrete Mathematics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

Evaluate the line integrals in Exercises 45–50.
$\int_{C} f (x, y, z) d s, where f (x, y, z) = (x + z) 3^{y}$ and C is the curve parameterized by role="math" localid="1651219517485" $x = 2 - t, y = 4 t, z = t + 5, for t = 1 to t = 4$ .