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.

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

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∫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

Torque and Rotational Motion

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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.)