Finding the volume of a shaded region above a graph?

The diagram shows part of the curve #y = x^2 + 1#. Find the volume obtained when the shaded region is rotated through #360°# about the #y#-axis.
The shaded regions boundaries are the following: #0 <= x <= 2# and #1 <= y <= 5#.
Graph sketched here

1 Answer
Jim G.
Sep 16, 2017

#V=8pi" cubic units"#

Explanation:

#"for volume (V) rotated about the y-axis"#

#•color(white)(x)V=piint_c^d x^2dy#

#"here "c=1" and "d=5#

#y=x^2+1rArrx^2=y-1#

#rArrV=piint_1^5(y-1)dy#

#color(white)(rArrV)=pi[1/2y^2-y]_1^5#

#color(white)(rArrV)=pi[25/2-5-(1/2-1)]#

#color(white)(rArrV)=8pi" cubic units"#

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