Chapter 16.4, Problem 3E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.

**3.** `∮`

_{C}` `

*x**y dx* + *x*^{2}*y*^{3} *dy*,

*C* is the triangle with vertices (0, 0), (1, 0), and (1, 2)

(a)

To determine

**To evaluate**: the line integral in direct method.

Explanation

**Given data:**

Line integral is
*C* is triangle with vertices

Draw the triangular curve with vertices

From Figure 1, write the expressions for curve

Differentiate the equation (1) with respect to *t*.

Differentiate the equation (2) with respect to *t*.

From Figure 1, write the expressions for curve

Differentiate the equation (3) with respect to *t*.

Differentiate the equation (4) with respect to *t*.

From Figure 1, write the expressions for curve

Differentiate the equation (5) with respect to *t*.

Differentiate the equation (6) with respect to *t*.

Find the value of line integral

**(b)**

To determine

**To evaluate**: The line integral using Green’s Theorem.

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