Chapter 6: Q5P (page 314)

$\int_{C} (y e^{x} - 1) d x + e^{x} d y$ ,where Cis the semicircle through $(0, - 10), (10, 0), a n d (0, 10)$ (Compare Problem 4.)

Short Answer

Expert verified

The solution to this problem is 20

Step by step solution

Given Information.

The given information is $\int_{C} (y e^{x} - 1) d x + e^{x} d y$ .

Definition of Green’s Theorem.

The Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

Find the solution.

Consider an equation and name it as equation 1

$\int_{C} (y e^{x} - 1) d x + e^{x} d y . . . . . . . (1)$

Compare equation (1) to Green’s Theorem.

$P = y e^{x} - 1 Q = e^{x}$

Find the differentiation with respect to.

$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0$

This implies that the surface integral will also be zero, which further leads to the line integral along C is negative that of line integral moving from

$(0, 10) t o (0, - 10) .$

Let the value be as follows:

$x = 0 d x = 0$

Find the Integral.

$l = \int_{10}^{- 10} d y = - (- 10 - 10) = 20$

Hence, the solution to this problem is 20.

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 Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$\int_{C} (y e^{x} - 1) d x + e^{x} d y$ ,where Cis the semicircle through $(0, - 10), (10, 0), a n d (0, 10)$ (Compare Problem 4.)

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Translational Dynamics

Rotational Dynamics

Energy Physics

Electricity

Measurements

Study anywhere. Anytime. Across all devices.

∫C(yex-1)dx+exdy,where Cis the semicircle through(0,-10),(10,0),and(0,10)(Compare Problem 4.)

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Translational Dynamics

Rotational Dynamics

Energy Physics

Electricity

Measurements

Study anywhere. Anytime. Across all devices.

$\int_{C} (y e^{x} - 1) d x + e^{x} d y$ ,where Cis the semicircle through $(0, - 10), (10, 0), a n d (0, 10)$ (Compare Problem 4.)