Question

Asked Feb 20, 2019

Find the coordinates of the centroids of the attached figure. Each region is covered by a thin plate.

The solid generated by revolving the region bounded by

y=x^3, y=0 and x=1 about the x axis

Step 1

You have asked two unrelated questions. Your first question is "to find the centroid of the attached figure that is a semi circle".

Your second question is "The solid generated by revolving the region bounded by y=x^3, y=0 and x=1 about the x axis". I will address your first question. Please post the second question **completely** as a separate question.

Step 2

Please see the attached figure. Since the figure is symmetrical about the y - axis, it\'s centroid C will be located on y axis. Hence, the x coordinate of the centroid is 0. We need to work out the y coordinate of the centroid.

A semi circle is made of several tiny plates of thickness dy. One such plate is shown as hashed area in the figure on the white board. The length of this plate = 2x; thickness = dy

Hence, Area of this plate, dA = Length x thickness = 2x.dy

Moment of this area at the centre of the circle (or origin) = A x distance from the origin = A x y = 2x.dy.y = 2xy.dy

Step 3

Equation of a circle is given by: x2 + y2 = a2;

Hence, x = square root of (a2 - x2)

Further range of integration to cover the entire semi circle with be y = 0 to y =a

Area of th...

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