Part II: The Answers
420
Answers
601–700
In this case, the cross-sectional slice has the shape of a washer. When the cross-sectional
slice is a washer, you can find A(x) by using
A
r
r
=
−
= ( ) − ( )
π
π
π
(
)
(
)
outer radius
inner radius
out
i n
2
2
2
2
π
where the outer radius, r out , is the distance of the function farther away from the line of
rotation and the inner radius, r in , is the distance of the function closer to the line of
rotation at a particular value of x.
On the interval 0 3
, π
( ) , you have sec x < 4. Because you’re rotating the region about the
line y = 4, you have r out = 4 and r in = 4 – sec x. Therefore, the integral to find the volume is
π
π
π
π
π
4
4
8
8
2
2
0
3
2
0
3
− −
=
−
(
)
=
+
∫
∫
(
sec )
sec
sec
ln sec
tan
x
dx
x
x dx
x
x − −
=
+
−
−
+
−
)
(
tan
ln sec
tan
tan
l n sec
tan
tan
x
0
3
8
3
3
3
8
0
0
0
π
π
π
π
π
=
+
−
(
)
8 2
3
3
ln
π
680.
832
15
π
Recall the definition of volume: Let S be a solid that lies between y = c and y = d. If A(y)
is the cross-sectional area of S in the plane P y that goes through y and is perpendicular
to the y-axis, where A is a continuous function, then the volume of S is
V
A y dx
c
d
= ∫ ( )
In this case, the cross-sectional slice has the shape of a washer. When the cross-sectional
slice is a washer, you can find A(y) by using
A
r
r
=
−
= ( ) − ( )
π
π
π
π
(
)
(
)
outer radius
inner radius
out
i n
2
2
2
2
420
Answers
601–700
In this case, the cross-sectional slice has the shape of a washer. When the cross-sectional
slice is a washer, you can find A(x) by using
A
r
r
=
−
= ( ) − ( )
π
π
π
(
)
(
)
outer radius
inner radius
out
i n
2
2
2
2
π
where the outer radius, r out , is the distance of the function farther away from the line of
rotation and the inner radius, r in , is the distance of the function closer to the line of
rotation at a particular value of x.
On the interval 0 3
, π
( ) , you have sec x < 4. Because you’re rotating the region about the
line y = 4, you have r out = 4 and r in = 4 – sec x. Therefore, the integral to find the volume is
π
π
π
π
π
4
4
8
8
2
2
0
3
2
0
3
− −
=
−
(
)
=
+
∫
∫
(
sec )
sec
sec
ln sec
tan
x
dx
x
x dx
x
x − −
=
+
−
−
+
−
)
(
tan
ln sec
tan
tan
l n sec
tan
tan
x
0
3
8
3
3
3
8
0
0
0
π
π
π
π
π
=
+
−
(
)
8 2
3
3
ln
π
680.
832
15
π
Recall the definition of volume: Let S be a solid that lies between y = c and y = d. If A(y)
is the cross-sectional area of S in the plane P y that goes through y and is perpendicular
to the y-axis, where A is a continuous function, then the volume of S is
V
A y dx
c
d
= ∫ ( )
In this case, the cross-sectional slice has the shape of a washer. When the cross-sectional
slice is a washer, you can find A(y) by using
A
r
r
=
−
= ( ) − ( )
π
π
π
π
(
)
(
)
outer radius
inner radius
out
i n
2
2
2
2