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Use cylindrical coordinates to evaluate d V when

Use cylindrical coordinates to evaluate d V when

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Use cylindrical coordinates to evaluate X d V R when Ri x'+y?<l, x20, y20, 05z E2
Transcribed Image Text:Use cylindrical coordinates to evaluate X d V R when Ri x'+y?<l, x20, y20, 05z E2
Expert Solution
Step 1

The given integral is ∭RxdV where the region R is defined by R:x2+y2≤1, x≥0, y≥0, 0≤z≤2.

Converting to cylindrical coordinates, we have

x=rcosθy=rsinθz=zdV=rdzdrdθx2+y2=r2cos2θ+r2sin2θ=r2cos2θ+sin2θ=r2

From the given definition of R, we have 0≤z≤2.

Also, x2+y2≤1⇒r2≤1.

Thus, 0≤r2≤1 and hence 0≤r≤1.

Note that, x≥0, y≥0 implies that the region lies in the first quadrant in the xy-plane.

Hence, the value 0≤θ≤π2.

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