Suppose that R is the finite region bounded by f(x)= √x and g(x)= x/3. Find the object's exact volume when rotating R about the x-axis. Find the exact volume when rotating R about the y-axis

Answer 1

Hi Sam,

Graphs of the two lines forming R intersect at x=9

Therefore , the volume V_x of the body formed by the R rotating around x-axis (

V_x=∫[0,9] π{(√x)²-(x/3)²dx}=∫[0,9] π{[x-x²/9]dx}=

π{x²/2-x³/27}_[0,9]

If R would be rotating around y-axis,

we can re-write the bordering lines as x=y² and x=3y (meeting at y=3).the volume V_y of the body formed by the R rotating around y-axis would be

V_y=∫[0,3] π{[(3y)²-y⁴]dy}= π{(y³-y⁵/5)}_[0,3]

The rest is to evaluate V_x at x=9 and V_y at y=3 b/c at lower ends [for x=0 V_x=0 and for y=0 V_y=0]

Hope it's helpful

Dr.Ariel B.

Answer 2