(a) Use calculus to find the circumference of a circle with radius
r
.
(b) Use calculus to sum the areas between concentric circles and find the area of a
circle. Hint: Show
dA
= 2
πx dx
(c) Use polar coordinates and double integrals to find the area of a circle.
345.
(a) Find common area of intersection associated with the
circles
x
2
+
y
2
=
r
2
0
and
(
x

r
0
)
2
+
y
2
=
r
2
0
(b) Find the volume of the solid of revolution if this area
is rotated about the
x

axis.
Hint: Make use of symmetry.
346.
Sketch the region
R
over which the integration is to be performed
(
a
)
1
0
y
y
2
f
(
x, y
)
dx dy
(
b
)
π
0
3 cos
θ
0
f
(
r, θ
)
r dr dθ
(
c
)
4
1
x
2
1
f
(
x, y
)
dy dx
347.
Sketch the region of integration, change the order of integration and evaluate
the integral.
(
a
)
3
1
2
y

1
12
xy dx dy
(
b
)
4
0
√
x
x/
2
3
xy dy dx
(
c
)
b
a
d
c
xy dx dy,
a
≤
x
≤
b
c
≤
y
≤
d
348.
Integrate the function
f
(
x, y
) =
3
2
xy
over the region
R
bounded by the curves
y
=
x
and
y
2
= 4
x
(a) Sketch the region of integration.
(b) Integrate with respect to
x
first and
y
second.
(c) Integrate with respect to
y
first and
x
second.
268
349.
Evaluate the double integral and sketch the region of integration.
I
=
1
0
x
0
2(
x
+
y
)
dy dx.
350.
For
f
(
x
) =
x
and
b > a >
0
, find the number
c
such that the mean value
theorem
b
a
f
(
x
)
dx
=
f
(
c
)(
b

a
)
is satisfied. Illustrate with a sketch the geometrical
interpretation of your result.
351.
Make appropriate substitutions and show
dx
α
2
+
x
2
=
1
α
tan

1
x
α
+
C
dx
α
2

x
2
=
1
α
tanh

1
x
α
+
C,
x < α
dx
b
2
+ (
x
+
a
)
2
=
1
b
tan

1
x
+
a
b
+
C
dx
b
2

(
x
+
a
)
2
=
1
b
tanh

1
x
+
a
b
+
C,
x
+
a < b
352.
(a) If
f
(
x
) =
f
(
x
+
T
)
for all values of
x
, show that
nT
0
f
(
x
)
dx
=
n
T
0
f
(
x
)
dx
(b) If
f
(
x
) =

f
(
T

x
)
for all values of
x
, show that
T
a
f
(
x
)
dx
=

a
0
f
(
x
)
dx
353.
(a)
Use integration by parts to show
e
ax
sin
bx dx
=
1
a
e
ax
sin
bx

b
a
e
ax
cos
bx dx
and
e
ax
cos
bx dx
=
1
a
e
ax
cos
bx
+
b
a
e
ax
sin
bx dx
(b) Show that
e
ax
sin
bx dx
=
e
ax
a
sin
bx

b
cos
bx
a
2
+
b
2
+
C
e
ax
cos
bx dx
=
e
ax
b
sin
bx
+
a
cos
bx
a
2
+
b
2
+
C
354.
Make an appropriate substitution to evaluate the given integrals
(
a
)
(
e
2
x
+ 3)
m
e
2
x
dx
(
b
)
e
4
x
+
e
3
x
e
x
+
e

x
dx
(
c
)
(
e
x
+ 1)
2
e
x
dx
355.
Evaluate the given integrals
(
a
)
x
+
a
x
3
dx
(
b
)
(
x
+
a
)(
x
+
b
)
x
3
dx
(
c
)
(
x
+
a
)(
x
+
b
)(
x
+
c
)
x
3
dx
356.
Evaluate the given integrals
(
a
)
ln(1 +
x
)
dx
(
b
)
x
4
+ 1
x

1
dx
(
c
)
(
a
+
bx
)(
x
+
c
)
m
dx
269
357.
Use a limiting process to evaluate the given integrals
(
a
)
∞
0
e

st
dt,
s >
0
(
b
)
x
∞
e
t
dt
(
c
)
∞
0
te

st
dt,
s >
0
358.
Consider a function
J
n
(
x
)
defined by an infinite series of terms and having
the representation
J
n
(
x
) =
x
n
2
n
1
n
!

x
2
2
2
1!(
n
+ 1)!
+
x
4
2
4
2!(
n
+ 2)!
+
· · ·
+
(

1)
m
x
2
m
2
2
m
m
!(
n
+
m
)!
+
· · ·
where
n
is a fixed integer and
m
represents the
m
th term of the series. Here
m
takes
on the values
m
= 0
,
1
,
2
, . . .
. Show that
x
0
J
1
(
x
)
dx
= 1

J
0
(
x
)
The function
J
n
(
x
)
is called the Bessel function of the first kind of order
n
.
359.
Determine a general integration formula for
I
n
=
x
n
e
x
dx
and then evaluate
the integral
I
4
=
x
4
e
x
dx
360.
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 Fall '15
 Derivative, Cartesian Coordinate System, Polar coordinate system