Question

4. Find the volume of the solid generated by revolving the region bounded by the given line and curve about the y-axis. x + y2 +1, x = 2

Answer 1

Solution:

The volume of the solid generated by revolving about the y-axis a region between the curves x = f(y) and x = g(y), where g(y)<f(y) and c <y<dis given by the formula

d V=T (If (y)]? – [g(y)]°) dy. с

yA d = 8 (y) = f(y) C 0 X

Given

x = y2 + 1, x = 2 revolved around the y-axis

The intersection point of two curves are (2, 1) and (2, -1).

Now using the above concept here x=2=f(y) and x=y^2+1=g(y) ranges from y=-1 to y=1 therefore the volume of revolution about y-axis

V=n ſ122 – (y2 + 1)2]dy -1

V = [4 - [4 – (y+1 + 2y)]dy -1

V = [3 – [3 – y4 – 2y2]dy - Los -1

ys 2y| 1 V = 3y - 5 3 -1

2 2 * 21 V = TT == [3(2) - - 3

41 V = [6-3 3)

190-6-20 V = TT 15 V =

V 64 TT 15