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Chapter 13: Q. 41 (page 1079)

Evaluate the double integrals in Exercises 39鈥�48. Use suitable transformations as necessary.

鈭�鈭�R2y-x3x+y+1dA, where R is the parallelogram with vertices (0, 0), (2, 1), (1, 4), and (鈭�1, 3) .

Short Answer

Expert verified

鈭�鈭�R2y-x3x+y+1dA=212ln(2)

Step by step solution

01

Draw the region

Plot the given points to form the region and name the vertices.

Consider the new set of variables defined as,

u=x-2yv=3x+y

From the above equations,

u+2v7=xv-3u7=y

Plot these limits on u v plane,

02

Find the integration

Use the antiderivative to evaluate the integrals,

鈭�鈭�R2y-x3x+y+1dA=-37ln(2)鈭�-70udu=-37ln(2)u22-70=-37ln(2)-492=212ln(2)

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

鈭�-33鈭�-9-x29-x2鈭�-9-x2-y29-x2-y2f(x,y,z)dzdydx

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral 鈭�ab鈭�cdf(x,y)dydx.

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.

鈭�鈭�Rx3ex2ydA,whereR={(x,y)|0鈮�x鈮�4and鈭�1鈮�y鈮�1}

Evaluate the triple integrals over the specified rectangular solid region.

鈭�R鈥�zsin鈦�xcos鈦�ydV,whereR=(x,y,z)鈭�0鈮�x鈮�蟺,3蟺2鈮�y鈮�2蟺,and1鈮�z鈮�3

In Exercises 45鈥�52, rewrite the indicated integral with the specified order of integration.

Exercise 41 with the order dy dx dz.
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