document.write( "Question 1172398: What is the volume of solid in xyz-space bounded by surfaces y = x^2, y = 2 - x^2, z = 0 and z = y + 3? \n" ); document.write( "

Algebra.Com's Answer #850840 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "What is the volume of solid in xyz-space bounded by surfaces y = x^2, y = 2 - x^2, z = 0 and z = y + 3?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I will solve this problem mentally.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "In (x,y)-plane, the area concluded between y = x^2, x-axis, and 0 <= x <= 1 is 1/3.\r\n" );
document.write( "\r\n" );
document.write( "It is elementary calculation from Calculus.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From it, we deduce, that the area concluded between y = x^2, y = 1  and 0 <= x <= 1 is 2/3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, the area concluded between y = x^2 and y = 2-x^2 is 4 times 2/3, or 8/3 square units.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now, our 3D solid consists of two parts.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "One part is a right cylinder 0 <= z <= 3 over its base, which the area concluded \r\n" );
document.write( "between y = x^2 and y = 2-x^2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The volume of this cylinder is   = 8 cubic units.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The other part is half of the cylinder  3 <= z <= 5  with the same base.\r\n" );
document.write( "\r\n" );
document.write( "The height of this imaginary cylinder is 2 units (z from 3 to 5), so, its volume is  = 16/3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The plane z = z + 3 cuts this cylinder in two parts of equal volumes - it is clear from the symmetry.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the whole volume of the 3D body under the interest is   =  =  = 32/3 cubic units.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved mentally.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " My interior voice tells me that this mental solution\r
\n" ); document.write( "\n" ); document.write( " is what is expected in this problem.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );