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| PAGE: 676 | SET: Exercises | PROBLEM: 36 |
(a)
The curves are 
and 
Substitute 
in the above equation
The direction of the closed loop is towards origin.
The limits of 
is 
Find area:
Area is region enclosed by the astroid equai to 
into area of the region in the first quadrant.
Area is region enclosed the - axis is 
.
Area is enclosed by the curve is 
.
(b)
Find volume:
The curves are and
Consider,
Differentiate on each side with respect to 
.
Change the limits
The volume of the solid is above - axis is 
.
Substitute and 
in the formula.
The volume of the solid is 
.
(c)
Find the centroid.
By symmetry, the 
-coordinate of the centroid is zero.
Replace 
.
Substitute .
Substitute 
in the above equation
The centriod is 
.
(a)Area is enclosed by the curve 
is 
(b) The volume of the solid is 
is 
(c) The centriod is 
.
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