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Chapter 13: Q 30 (page 1066)

Describe the graphs of the equations in and provide alternative equations in the specified coordinate systems.

Changez=x+y to the cylindrical and spherical systems.

Short Answer

Expert verified

The cylindrical equation isr=zcos胃+sin胃

And the spherical equation iscot蠒=cos胃+cos蠒

Step by step solution

01

Step 1:Given information

The given expression isz=x+y

02

Step 2:Simplification

Consider the equation z=x+y

To change z=x+yinto cylindrical system,we can use

x=rcos胃,y=rsin胃

so,

z=x+y

鈬�z=rcos胃+rsin胃

鈬�r=zcos胃+sin胃

Therefore the cylindrical system is r=zcos胃+sin胃

Now,

to change z=x+yinto spherical system,we can use

x=蚁sin蠒cos胃,y=蚁sin蠒cos蠒,z=cos蠒

so,

z=x+y

鈬�蚁cos蠒=蚁sin蠒cos胃+蚁sin蠒cos蠒

鈬�蚁cos蠒=蚁sin蠒cos胃+cos蠒

鈬�蚁cos蠒蚁sin蠒=cos胃+cos蠒

鈬�cot蠒=cos胃+cos蠒

Therefore the equation in spherical system iscot蠒=cos胃+cos蠒

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Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

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