415
Answers
601–700
Answers and Explanations
674.
π
π
2
8 4
+
Recall the definition of volume: Let S be a solid that lies between x = a and x = b. If A(x)
is the cross-sectional area of S in the plane P x that goes through x and is perpendicular
to the x-axis, where A is a continuous function, then the volume of S is
V
A x dx
a
b
= ∫ ( )
Here’s the region that’s being rotated. Because the cross-sectional slice is a circle, you
have A(x) = π(y)
2
= π(f
 
(x))
2
.
Because the function is bounded by x = 0 and x = 1, these are the limits of integration.
With y
x
= +
1
1
2
, the integral to find the volume becomes
π
π
1
1
1
1
2
2
2
2
0
1
0
1
+






=
+
(
)
∫
∫
x
dx
x
dx
Now use the trigonometric substitution x = tan θ so that dx = sec
2 
θ
 
dθ. Find the new limits
of integration by noting that if x = 1, then 1 = tan θ so that π
4
= θ . Likewise, if x = 0, then
0 = tan θ so that 0 = θ. Using these new values, you produce the following integral:
π
θ
θ
θ π
θ
θ
θ
π
θ θ
π
π
π
sec
tan
sec
sec
cos
2
2
2
0
4
2
2
2
0
4
2
0
4
1 +
(
)
=
(
)
=
∫
∫
∫
d
d
d
Now use the identity cos
2
θ =
1
2
1
2
( cos( ))
+
θ :
π
θ θ π θ
θ
π π
π
π
π
1
2
1
2
2
2
2
2 4
2
2
0 0
0
4
0
4
( cos( ))
sin
sin
(
)
+
=
+
(
)
=
+
− +




∫
d
 





=
+
π
π
2
8 4