Chapter 14: Q 19. (page 1106)

In Chapter 4 we defined the average value of an integrable function $f$ on the interval $[a, b]$ to be $\frac{1}{b - a} \int_{a}^{b} f (x) d x$ .
Generalize this expression to a formula that will give the average value of $f (x, y, z)$ along a smooth curve $C$ parametrized by $r (t)$ for $a \leq t \leq b$ .

Short Answer

Expert verified

The parametrized equation is $f_{a v g} (C) = \frac{\int_{C} f (x, y, z) d s}{\int_{C} d s}$ .

Step by step solution

Given Information

It is given that $f_{avg} = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

We need to generalize the expression to find average value of $f (x, y, z)$ along a smooth curve parametrized by $r (t)$ for $a \leq t \leq b$ .

Simplifying average value of function

Since $f_{a v g} = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

$= \frac{\int_{a}^{b} f (x) d x}{b - a}$

$= \frac{\int_{a}^{b} f (x) d x}{\int_{a}^{b} d x}$

Deducing the formula for average value

The value of integral is along smooth curve is

$\int_{C} f (x, y, z) d s$

The average value is

role="math" localid="1653588021219" $f_{a v g} (C) = \frac{\int_{C} f (x, y, z) d s}{Arc Length of C for a \leq t \leq b}$

$= \frac{\int_{C} f (x, y, z) d s}{\int_{C} d s}$

Hence, the formula for average value of $f (x, y, z)$ along smooth curve parametrized by $r (t)$ for $a \leq t \leq b$ is

$f_{a v g} (C) = \frac{\int_{C} f (x, y, z) d s}{\int_{C} d s}$

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.

Short Answer

Step by step solution

Given Information

Simplifying average value of function

Deducing the formula for average value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Calculus

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Chapter 4 we defined the average value of an integrable function fon the interval [a,b]to be 1b-a∫abf(x)dx. Generalize this expression to a formula that will give the average value of f(x,y,z) along a smooth curve Cparametrized by r(t) fora≤t≤b.

Short Answer

Step by step solution

Given Information

Simplifying average value of function

Deducing the formula for average value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Calculus

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.