Chapter 16.4, Problem 4E

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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.

**4.** `∮`

_{C}*x*^{2}*y*^{2} *dx* + *xy dy*, *C* consists of the arc of the parabola *y* = *x*^{2} from (0, 0) to (1, 1) and the line segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0)

**(a)**

To determine

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

Explanation

**Given data:**

Line integral is
*C* comprises of parabola from

Draw the curve *C* as shown in Figure 1.

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*

**(b)**

To determine

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

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