MATH 106A - CALCULUS II
FALL 2005
QUIZ 3
NAME:
Show ALL your work CAREFULLY.
(a) Sketch the region bounded by the curve y+1=x2and the line y=x+1.
(2,3)
(b) Set up (without computing the actual integral) the definite integral repre-
senting the volume of the solid of revolution formed when the region in (a) is
revolved around the line y=โ1.
A typical slice (shown in the figure) when revolved around y=โ1is a
washer with surface area ฯ[R2โr2]where Ris the distance from the line
y=x+1 to the line y=โ1and ris the distance from the curve y=x2โ1
to the line y=โ1. Therefore, the volume of the solid of revolution is
given by
V=๎2
โ1
ฯ[(x+2)
2
โx4]dx =ฯ๎2
โ1
4+4x+x2
โx4dx.
(c) Find the area of the region in (a). [first set it up and then evaluate]
Date: September 26, 2005.
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