Use single integration (use a thin, long vertical sliceas your integration element) to determine the x and y coordinates of thecentroid of the plane region shown.y (mm)xy = 3175x (mm)102 mm25 mm127 mm

Answered: Use single integration (use a thin,…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Use single integration (use a thin, long vertical slice
as your integration element) to determine the x and y coordinates of the
centroid of the plane region shown.
y (mm)
xy = 3175
x (mm)
102 mm
25 mm
127 mm

Expert Solution

Step 1 Area of the whole region.

XY = 3175

Y = 3175 / X

Taking a vertical strip of area dA,

dA = Y . dx

dA = [ 3175 / X ] dx

Area = $\int d A$

$= \int_{25}^{120} \frac{3175}{X} d x$

$= 3175 [\ln (120) - \ln (25)]$

Area $= 4980.355 m m^{2}$

Step 2 Y coordinate of the centroid.

The centroid coordinates are given by,

$\bar{y} = \frac{\int Y d A}{A} \bar{x} = \frac{\int X d A}{A}$

For y coordinate,

$\bar{y} = \frac{\int Y d A}{A}$

$\bar{y} = \frac{\int_{25}^{120} \frac{3175}{2 X} [\frac{3175}{X} d x]}{A}$

$= \frac{\frac{{(3175)}^{2}}{2} {[\frac{1}{X}]}_{25}^{120}}{A}$

$= \frac{- 5040312.5 [\frac{1}{120} - \frac{1}{25}]}{A} = \frac{159609.8958}{4980.335}$

$\bar{y} = 32.04 m m$

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Use single integration (use a thin, long vertical slice as your integration element) to determine the x and y coordinates of the centroid of the plane region shown. y (mm) xy = 3175 x (mm) 102 mm 25 mm 127 mm