Chapter 16, Problem 15RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Verify that Green’s Theorem is true for the line integral ∫_{C} xy^{2} *dx* − *x*^{2}*y dy*, where *C* consists of the parabola *y = x*^{2} from (-1, 1) to (1, 1) and the line segment from (1, 1) to (-1, 1).

To determine

**To Verify:** TheGreen’s Theorem is true for the line integral
*C* consists of the parabola

Explanation

**Given data:**

*C* consists of the parabola

**Formula used:**

Write the expression Greenâ€™s Theorem.

Find the value of limits as follows.

Draw the region

Figure 1, consists of two parts as follows.

Parabolic function represented as folows.

Line function represented as follows.

Write the expression for

Consider the expression as follows.

Modify equation (5) as follows.

Compare equations (2) and (4).

Differentiate equation (7) with respect to

Differentiate equation (8) with respect to

Find the value of

Substitute

Integrate and apply limits as follows.

Simplify expression as follows.

Compare equations (3) and (4).

Differentiate equation (9) with respect to

Differentiate equation (10) with respect to

Find the value of

Substitute

Integrate and apply limits as follows.

Simplify expression as follows

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