Answer to Question #307283 in Civil and Environmental Engineering for isaiah

Question

Calculate the volume of the solid formed by revolving about the line y=1 the region bounded by the parabola

x

2 = 4y and that line. Take the rectangular elements of area parallel to the axis of revolution.

Answer 1

Point of intersection of y=1

Since x²=4y, then; y=1 and x=2

Hence integration is to be carried out be between x=2 and y=1

Line is above the curve

Taking vertical strips of width dx and rotating about x-axis we the volume generated as

dV= πy²(line)-πy²(curve)=π{(2x)²-(x²)²}dx

or dV=π(2x²-x⁴)dx

Hence Volume V=π∫(2x²-x⁴)

=π(2x³/2-x⁵/2)

=π[{(2×1³/2–1⁵/1)-(2×2³/2–1⁵/1)}]

=π(2/1–7/1)

=π(9.429/1)=9.429π cubic units