Answer :
V = 19.739 is the volume of the solid below the surface of revolution and above the xy-plane.
What is the solid's volume?
The amount of space that a solid occupies is expressed in terms of its volume. It is calculated using the quantity of unit cubes required to completely fill the solid.
We have 30 unit cubes in the solid after counting the unit cubes, hence the volume is 2 units. 3 units 30 cubic units from 5 units.
Given Curve
z = sin(y) , (0 ≤ Y ≤ π)
It's on the y-z plane to see the volume of revolution.
It is the upper portion of the sine curve, from 0 to π, revolved around the z - axis.
A differential area element is then dA = z dx. Revolving this around the z - axis gives the differential volume element,
dV = 2πr dA
dV = 2π (z dx)
dV = 2πy sin(y) dx
Then, integrating (I used integration by parts, u = y, dv = sin(y) dy)
V = 2π{ - y cos(y) + sin(y)} evaluated between π and 0
V = 2π{ ( - π cos(π) + sin(π)) - ( - 0*cos(0) + sin(0)) }
V = 2π²
Then, to 3 decimal places
V = 2(3.14159265)2
V = 19.739
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