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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes.

12. R is parallelogram with vertices (0, 0), (4, 3), (2, 4), (–2, 1)

To determine

To find: The equation for the transformation T maps with given rectangular region S.

Explanation

Given:

A region R is the parallelogram with vertices (0,0),(4,3),(2,4) and (2,1) .

Calculation:

Obtain the line by using line equation xx1x2x1=yy1y2y1 .

Compute the equation of the line that passes through the vertices (0,0) and (4,3) .

x040=y030x4=y33x=4y3x4y=0

Thus, the equation of the line that passes through the vertices (0,0) and (4,3) is 3x4y=0 .

Compute the equation of the line that passes through the vertices (4,3) and (2,4) .

x424=y343x42=y311(x4)=2(y3)

Simplify further as follows,

x4=2y+6x+2y=6+4x+2y=10

Thus, the equation of the line that passes through the vertices (4,3) and (2,4) is x+2y=10 .

Compute the equation of the line that passes through the vertices (2,4) and (2,1)

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