calc_tutorial_17_1_004

 
Problem Statement

{2,3,4,5,6,7,8}
3+1
3+4
round(4/(pow(3,2)+3*4),3)

Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise.

, where C is the triangle with vertices (0, 0), (1, 0), (1, 1).

 
Step 1

Question Sequence

Question 1

Recall how to apply Green's Theorem. Let D be a domain whose boundary ∂D (diffD) is a simple closed curve, oriented counterclockwise. Then we have the following equation.

In the figure above, we see that ∂D = C is the triangle bounding the shaded region. In addition, the triangle is oriented counterclockwise as indicated in the problem statement.

Because C = ∂D, we note that is already in the correct form with

F1 =

A.
B.
C.
D.

and

F2 =

A.
B.
C.
D.
Correct.
Incorrect.

 
Step 2

Question Sequence

Question 2

To apply Green's Theorem to solve , we will need to calculate the appropriate partial derivatives and find the appropriate bounds of integration for the double integral regarding domain of the triangle.

Calculate the partial derivatives (∂F2 / ∂x) and (∂F1 / ∂y) where F1 = and F2 = .

(∂F2 / ∂x) =

(∂F1 / ∂y) =

Therefore we have (∂F2 / ∂x) − (∂F1 / ∂y) =

Correct.
Incorrect.

 
Step 3

Question Sequence

Question 3

Find the appropriate bounds of integration for the double integral.

Notice that the triangle is bounded above by a segment of the line y = x and bounded below by the x-axis between x = 0 and x = 1. We also see that the height of the region, y, is between 0 and x as x increases from 0 to 1.

With this information, use Green's Theorem to set up the appropriate double integral.

Correct.
Incorrect.

 
Step 4

Question Sequence

Question 4

Use Green's Theorem to evaluate the line integral. Round your answer to three decimal places.

Correct.
Incorrect.