calculus test bank

8.6. POWER SERIES 349

The set A is bounded from above by q. Hence A has a least upper bound, say
R. Clearly |p| ≤ R < q and hence R is a positive real number. Furthermore,

∞

cnxn converges for all x such that |x| < R and diverges for all x such that

n=0

|x| > R. We define R to be 0 for case (i) and R to be ∞ for case (ii). This
completes the proof of Theorem 8.6.3.

∞

Theorem 8.6.4 Let cn(x − a)n be any power series. Then exactly one

n=0

of the following three cases is true:

(i) The series converges only for x = a and the radius of convergence is 0.

(ii) The series converges for all x and the radius of convergence is ∞.

(iii) There exists a number R such that the series converges for all x such
that |x − a| < R and diverges for all x such that |x − a| > R.

∞

Proof. Let u = x − a and use Theorem 8.6.3 on the series cnun. The

n=0

details of the proof are left as an exercise.

∞

Theorem 8.6.5 If R > 0 and the series cnrn converges for |x| < R, then

n=0
∞∞

the series ncnxn−1, obtained by term-by-term differentiation of cnxn,

n=1 n=0

converges absolutely for |x| < R.

Proof. For each x such that |x| < R, choose a number r such that |x| < r <
∞

R. Then cnxn converges, lim cn rn = 0 and hence {cnrn}n∞=0 is bounded.

n=0 n→∞

There exists some M such that |cnrn| ≤ M for each natural number n. Then

∞ ∞ 1 x n−1
r r
|ncnxn−1| = n|cnrn| · ·

n=1 n=1

≤M ∞ x n−1
n.
rr
n=1

350 CHAPTER 8. INFINITE SERIES

∞ x n−1 x
The series n converges by the ratio test, since < 1. It follows
rr
n=1
∞

that ncnxn−1 converges absolutely for all x such that |x| < R. This

n=1

completes the proof of this theorem.

∞

Theorem 8.6.6 If R > 0 and the series cn(x − a)n converges for all x

n=0
∞

such that |x−a| < R, then the series cn(x−a)n may be differentiated with

n=0

respect to x any number of times and each of the differential series converges
for all x such that |x − a| < R.

∞

Proof. Let u = x−a. Then cnun converges for all u such that |u| < R. By

n=0
∞

Theorem 8.6.5, the series ncnun−1 converges for all u such that |u| < R.

n=1

This term-by-term differentiation process may be repeated any number of
times without changing the radius of convergence. This completes the proof
of this theorem.

Theorem 8.6.7 Suppose that R > 0 and f (x) = ∞ cnxn and R is radius
of convergence of the series n=0 continuous for all x
∞ xn
n=0 cn . Then f (x) is

such that |x| < R.

Proof. For each number c such that −R < c < R, we have

f (x) − f (c) ∞ xn − cn
x − c = cn x − c

n=0

∞

= cn nann−1

n=1
∞

≤ n cnann−1

n=1

8.6. POWER SERIES 351

for some an between c and x, for each natural number n, by the Mean Value
∞

Theorem. By Theorem 8.6.6, the series n |cnan|n−1 converges. Hence,

n=1

∞

lim |f (x) − f (c)| = lim |x − c| |c0 − c| + n cnann−1
x→c x→c
n=1

∞

= 0 · |c0 − c| + n cnann−1

n=1

= 0.

Hence, f (x) is continuous at each number c such that −R < c < R. This
completes the proof of this theorem.

∞

Theorem 8.6.8 Suppose that R > 0, f (x) = cnxn and R is the radius

n=0
∞

of convergence of the series cnxn. For each x such that |x| < R, we define

n=0

x

F (x) = f (t)dt.

0

Then, for each x such that |x| < R, we get

∞ xn+1
F (x) = cn n + 1 .

n=0

352 CHAPTER 8. INFINITE SERIES

Proof. Suppose that |x| < |r| < R. Then

m xn+1 x mx

lim F (x) − cn n + 1 = lim f (t)dt − cn tn
n→∞
m→∞ n=0 0 n=0 0

xm

= lim f (t) − cntn dt
m→∞ 0
n=0

x∞

= lim cntn dt
m→∞ 0
n=m+1

x ∞ dt

≤ lim |cntn|
m→∞ 0
n=m+1

x ∞ dt

≤ lim |cnrn|
m→∞ 0
n=m+1

≤ lim ∞ x
m→∞
|cnrn| 1 dt

n=m+1 0

= 0 · |x|

= 0,

∞

since |cnrn| converges.

n=0

It follows that

x x∞

f (t)dt = cntn dt

0 0 n=0

∞ xn+1
= cn n + 1 .

n=0

This completes the proof of the this theorem.

∞

Theorem 8.6.9 Suppose that f (x) = cnxn for all |x| < R, where R > 0

n=0
∞

is the radius of convergence of the series cnxn. Then f (x) has continuous

n=0

8.6. POWER SERIES 353

derivatives of all orders for |x| < R that are obtained by successive term-by-
∞

term differentiations of cnxn.

n=0

Proof. For each |x| < R, we define

∞

g(x) = ncnxn−1.

n=1

∞

Then, by Theorem 8.6.5, R is the radius of convergence of the series ncnxn−1.

n=1

By Theorem 8.6.7, g(x) is continuous. Hence,

x∞

c0 + g(x)dx = c0 + cnxn = f (x).

0 n=1

By the fundamental theorem of calculus, f (x) = g(x). This completes the
proof of this theorem.

Definition 8.6.2 The radius of convergence of the power series

∞

ak(x − a)k

k=1

is

(a) zero, if the series converges only for x = a;

(b) r, if the series converges absolutely for all x such that |x − a| < r and
diverges for all x such that |x − a| > r.

(c) ∞, if the series converges absolutely for all real number x.

If the radius of convergence of the power series in (x − a) is r, 0 < r < ∞,

then the interval of convergence of the series is (a − r, a + r). The end points

x = a + r or x = a − r are included in the interval of convergence if the
∞ ∞
corresponding series k=1 ak rk or k=1 (−1)kak rk converges, respectively. If

r = ∞, then the interval of convergence is (−∞, ∞).

354 CHAPTER 8. INFINITE SERIES

Exercises 8.4 In problem 1–12, determine the Taylor series expansion for
each function f about the given value of a.

1. f (x) = e−2x, a = 0 2. f (x) = cos(3x), a = 0

3. f (x) = ln(x), a = 1 4. f (x) = (1 + x)−2, a = 0

5. f (x) = (1 + x)−3/2, a = 0 6. f (x) = ex, a = 2

π π
7. f (x) = sin x, a = 8. f (x) = cos x, a =

6 4
π 10. f (x) = x1/3, a = 8
9. f (x) = sin x, a =
3

11. f (x) = sin x − 1 , a = 0 12. f (x) = cos x − 1 , a = 0
2 2

In problems 13-20, determine n f (k)(a) (x − a)k .
k!

k=0

13. f (x) = ex2, a = 0, n = 3 14. f (x) = x2e−x, a = 0, n = 3

1 16. f (x) = arctan x, a = 0, n = 3
15. f (x) = 1 − x2 , a = 0, n = 2 18. f (x) = arcsin x, a = 0, n = 3
17. f (x) = e2x cos 3x, a = 0, n = 4

19. f (x) = tan x, a = 0, n = 3 20. f (x) = (1 + x)1/2, a = 0, n = 5

8.7 Taylor Polynomials and Series

Theorem 8.7.1 (Taylor’s Theorem) Suppose that f, f , · · · , f (n+1) are all
continuous for all x such that |x − a| < R. Then there exists some c between
a and x such that

f (x) = Pn(x) + Rn(x)

where

n f (k)(a) (x − a)k , (x − a)n+1
k! (n + 1)!
Pn(x) = Rn(x) = f (n+1)(c) .

k=0

8.7. TAYLOR POLYNOMIALS AND SERIES 355

The polynomial Pn(x) is called the nth degree Taylor polynomial approxima-
tion of f . The term Rn(x) is called the Lagrange form of the remainder.

Proof. We define a function g of a variable z such that

g(z) = [f (x) − f (z)] − f (z)(x − z) − f (z)(x − z)2 − · · ·
1! 2!

− f (n)(z)(x − z)n − (x − z)n+1
n! Rn(x) (x − .

a)n+1

Then n f (k)(a)
and k!
g(a) = f (x) − (x − a)k + Rn(x) = 0,

k=0

g(x) = f (x) − f (x) = 0.

By the Mean Value Theorem for derivatives there exists some c between a
and x such that g (c) = 0. But

g (z) = −f (z) − [−f (z) + f (z)(x − z)] − −f (z)(x − z) + f (z)(x − z)2 −···
2!

− f n(z)(x − z)n−1 + f (n+1)(z)(x − z)n (n + 1)(x − z)n
− + Rn(x) (x − a)n+1
n! n!

= −f (n+1)(z) (x − z)n + (n + 1)(x − z)n
n! Rn(x) (x − a)n+1

g (c) = 0 = −f (n+1)(c) (x − c)n + (n + 1)(x − c)n
n! Rn(x) (x − a)n+1 .

Therefore,

Rn(x) = (x − a)n+1 · f (n+1)(c) = f (n+1)(c) (x − a)n+1
n+1 n! (n + 1)!

as required. This completes the proof of this theorem.

Theorem 8.7.2 (Binomial Series) If m is a real number and |x| < 1, then

(1 + x)m = 1 + ∞ m(m − 1) · · · (m − k + 1) xk
k!

k=1

= 1 + mx + m(m − 1) x2 + m(m − 1)(m − 2) x3 + · · · .
2! 3!

356 CHAPTER 8. INFINITE SERIES

This series is called the binomial series. If we use the notation

m m(m − 1) · · · (m − k + 1)
=

k k!

m
then is called the binomial coefficient and

k

∞ m xk.
k
(1 + x)m = 1 +

k=1

If m is a natural number, then we get the binomial expansion

m m xk.
k
(1 + x)m = 1 +

k=1

Proof. Let f (x) = (1 + x)m. Then for all natural numbers n,

f (x) = m(1 + x)m−1, f (x) = m(m − 1)(1 + x)m−2, · · · ,
f (n)(x) = m(m − 1) · · · (m − n + 1)(1 + x)m−n.

Thus, f (n)(0) = m(m − 1) · · · (m − n + 1), and

f (x) = ∞ m(m − 1)(m − 2) · · · (m − n + 1) xn
n!

n=0

∞ m xn
n
=

n=0

where m = 1 and m = m(m − 1) · · · (m − n + 1) is called the nth
0n

binomial coefficient. By the ratio test we get

lim m(m − 1) · · · (m − n)xn+1 · n!
(n + 1)! 1) · · · (m
n→∞ m(m − − n + 1)xn

= |x| lim m − n
n→∞ n + 1

= |x| lim m − 1
n
n→∞
1 + 1
n

= |x|,

8.7. TAYLOR POLYNOMIALS AND SERIES 357

and, hence, the series converges for |x| < 1.
This completes the proof of the theorem.

Theorem 8.7.3 The following power series expansions of functions are valid.

∞∞

1. (1 − x)−1 = 1 + xk and (1 + x)−1 = 1 + (−1)kxk, |x| < 1.

k=1 k=1

2. ∞ xk , e−x = 1 + ∞ xk , |x| < ∞.

ex = 1 + (−1)k
k! k!
k=1 k=1

∞ x2k+1 , |x| < ∞.

3. sin x = (−1)k (2k + 1)!

k=0

4. ∞ x2k , |x| < ∞.

cos x = (−1)k (2k)!

k=0

∞ x2k−1 , |x| < ∞.
(2k + 1)!
5. sinh x =

k=0

∞ x2k , |x| < ∞.
(2k)!
6. cosh x =

k=0

7. ln(1 + x) = ∞ xk+1 , −1 < x ≤ 1.

(−1)k k+1

k=0

1 1 + x ∞ x2k+1
8. ln = 2k + 1 , −1 < x < 1.
2 1−x
k=0

9. ∞ x2k+1 , −1 ≤ x ≤ 1.

arctan x = (−1)k 2k + 1

k=0

∞ −1/2 (−1)k x2k+1 , |x| ≤ 1.
k 2k + 1
10. arcsin x =

k=0

358 CHAPTER 8. INFINITE SERIES

Proof.
Part 1. By the geometric series expansion, for all |x| < 1, we have

1 ∞ 1 1 ∞
1−x =1+ 1+x 1 − (−x) = 1 +
xk and = (−1)kxk.

k=1 k=1

Part 2. If f (x) = ex, then f (n)(x) = ex and f (n)(0) = 1 for each n =

0, 1, 2, · · · . Thus ∞

ex = xn
.
n!
n=0

By the ratio test the series converges for all x.

lim xn+1 · n! = |x| lim 1
n→∞ (n + 1)! xn = 0.
n→∞ n + 1

Part 3. Let f (x) = sin x. Then f (x) = cos x, f (x) = − sin x, f (3)(x) =
− cos x and f (4)(x) = sin x. It follows that, for each n = 0, 1, 2, 3, · · · , we
have

f (4n)(0) = 0, f (4n+1)(0) = 1, f (4n+2)(0) = 0 and f (4n+3)(0) = −1.

Hence,

sin x = x − x3 + x5 − · · ·
3! 5!

∞ x2n+1
.
= (−1)n (2n + 1)!

n=0

By the ratio test, the series converges for all |x| < ∞:

lim (−1)n+1 x2n+3 (2n + 1)!
(2n + 3)! x2n+1
n→∞

= x2 lim 1

n→∞ (2n + 3)(2n + 2)

= 0.

Part 4. By term-by-term differentiation we get

cos x = (sin x) = ∞ x2n , |x| < ∞.
(2n)!
(−1)n

n=0

8.7. TAYLOR POLYNOMIALS AND SERIES 359

Part 5. For all |x| < ∞, we get

sinh x = 1 (ex − e−x)
2

1 ∞ xn − ∞ (−1)n xn
=
2 n! n!
n=0 n=0

∞ x2n+1
=.

(2n + 1)!

n=0

Part 6. By differentiating term-by-term, we get

∞ x2n |x| < ∞.
,l
cosh x = (sinh x) = (2n)!

n=0

Part 7. For each |x| < 1, by performing term by integration, we get

ln(1 + x) = x1 dx

0 1+x

∞∞ dx

= (−1)nxn

0 n=0

= ∞ xn+1
.
(−1)n n+1

n=0

Part 8. By Part 7, for all |x| < 1, we get

1 1+x = 1 [ln(1 + x) − ln(1 − x)]
ln 1−x 2

2

1 ∞ xn+1 − ∞ (−1)n (−x)n+1
=
(−1)n
2 n+1 n+1
n=0 n=0

1 ∞ (−1)n (1 − (−1)n+1)xn+1
=
2 n+1
n=0

∞ x2k+1
=.

2k + 1

k=0

1 1+x .
Recall that arctanh x = ln
2 1−x

360 CHAPTER 8. INFINITE SERIES

Part 9. For each |x| ≤ 1, we perform term-by-term integration to get

x1
arctan x = 0 1 + x2 dx

x∞

= (−1)kx2k dx

0 k=0

∞ x2k+1
.
= (−1)k (2k + 1)

k=0

Part 10. By performing term-by-term integration of the binomial series, we
get

arcsin x = x 1 dx
=
√
0 1 − x2

x

(1 − x2)−1/2dx

0

= x ∞ −1/2 (−x2)k dx
k
0 k=0

∞ −1/2 (−1)k x2k+1
.
= k (2k + 1)

k=0

This series converges for all |x| ≤ 1.
This completes the proof of this theorem.

8.8 Applications

Chapter 9

Analytic Geometry and Polar
Coordinates

A double right-circular cone is obtained by rotating a line about a fixed axis
such that the line intersects the axis and makes the same angle with the
axis. The intersection point of the line and the axis is called a vertex. A
conic section is the intersection of a plane and the double cone. Some of
the important conic sections are the following: parabola, circle, ellipse and a
hyperbola.

9.1 Parabola

Definition 9.1.1 A parabola is the set of all points in the plane that are
equidistant from a given point, called the focus, and a given line called the
directrix. A line that passes through the focus and is perpendicular to the
directrix is called the axis of the parabola. The intersection of the axis with
the parabola is called the vertex.

Theorem 9.1.1 Suppose that v(h, k) is the vertex and the line x = h − p is
the directrix of a parabola. Then the focus is F (h + p, k) and the axis is the
horizontal line with equation y = k. The equation of the parabola is

(y − k)2 = 4p(x − h).

361

362CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

Theorem 9.1.2 Suppose that v(h, k) is the vertex and the line y = k − p is
the directrix of a parabola. Then the focus is F (h, k + p) and the axis is the
vertical line with equation x = h. The equation of the parabola is

(x − h)2 = 4p(y − k).

9.2 Ellipse

Definition 9.2.1 An ellipse is the locus of all points, the sum of whose
distances from two fixed points, called foci, is a fixed positive constant that
is greater than the distance between the foci. The midpoint of the line
segment joining the two foci is called the center. The line segment through
the foci and with end points on the ellipse is called the major axis. The
line segment, through the center, that has end points on the ellipse and is
perpendicular to the major axis is called the minor axis. The intersections
of the major and minor axes with the ellipse are called the vertices.

Theorem 9.2.1 Let an ellipse have center at (h, k), foci at (h ± c, k), ends
of the major axis at (h ± a, k) and ends of the minor axis at (h, k ± b), where
a > 0, b > 0, c > 0 and a2 = b2 + c2. Then the equation of the ellipse is

(x − h)2 (y − k)2
+ = 1.

a2 b2

The length of the major axis is 2a and the length of the minor axis is 2b.

Theorem 9.2.2 Let an ellipse have center at (h, k), foci at (h, k ± c), ends
of the major axis at (h, k ± a), and the ends of the minor axis at (h ± b, k),
where a > 0, b > 0, c > 0 and a2 = b2 + c2. Then the equation of the ellipse
is

(y − k)2 (x − h)2
a2 + b2 = 1.

The length of the major axis is 2a and the length of the minor axis is 2b.

Remark 24 If c = 0, then a = b, foci coincide with the center and the
ellipse reduces to a circle.

9.3. HYPERBOLA 363

9.3 Hyperbola

Definition 9.3.1 A hyperbola is the locus of all points, the difference of
whose distances from two fixed points, called foci, is a fixed positive constant
that is less than the distance between the foci. The mid point of the line
segment joining the two foci is called the center. The line segment, through
the foci, and with end points on the hyperbola is called the major axis. The
end points of the major axis are called the vertices.

Theorem 9.3.1 Let a hyperbola have center√at (h, k), foci at (h ± c, k),
vertices at (h ± a, k), where 0 < a < c, b = c2 − a2, then the equation of

the hyperbola is

(x − h)2 − (y − k)2 = 1.
a2 b2

Theorem 9.3.2 Let a hyperbola have center√at (h, k), foci at (h, k ± c),
vertices at (h, k ± a), where 0 < a < c, b = c2 − a2, then the equation of

the hyperbola is

(y − k)2 − (x − h)2 = 1.
a2 b2

9.4 Second-Degree Equations

Definition 9.4.1 The transformations

x = x cos θ − y sin θ
y = x sin θ + y cos θ

and
x = x cos θ + y sin θ
y = −x sin θ + y cos θ

are called rotations. The point P (x, y) has coordinates (x , y ) in an x y -
coordinate system obtained by rotating the xy-coordinate system by an angle
θ.

Theorem 9.4.1 Consider the equation ax2 + bxy + cy2 + dx + ey + f =
0, b = 0. Let cot 2θ = (a − c)/b and x y -coordinate system be obtained

364CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

through rotating the xy-coordinate system through the angle θ. Then the
given second degree equation

ax2 + bxy + cy2 + dx + ey + f = 0

becomes ax2+cy2+dx+ey+f =0
where

a = a cos2 θ + b cos θ sin θ + c sin2 θ
c = a sin2 θ − b sin θ cos θ + c cos2 θ
d = d cos θ + e sin θ
e = −d sin θ + e cos θ
f =f

Furthermore, the given second degree equation represents
(i) an ellipse, a circle, a point or no graph if b2 − 4ac < 0;
(ii) a hyperbolic or a pair of intersecting lines if b2 − 4ac > 0;
(iii) a parabola, a line, a pair of parallel lines, or else no graph if b2 −4ac = 0.

9.5 Polar Coordinates

Definition 9.5.1 Each point P (x, y) in the xy-coordinate plane is assigned
the polar coordinates (r, θ) that satisfy the following relations:

x2 + y2 = r2, y = r cos θ, y = r sin θ.

The origin is called the pole and the positive x-axis is called the polar axis.
The number r is called the radial coordinate and the angle θ is called the
angular coordinates. The polar coordinates of a point are not unique as the
rectangular coordinates are. In particular,

(r, θ) ≡ (r, θ + 2nπ) ≡ (−r, θ + (2m + 1)π)

where n and m are any integers. There does exist a unique polar represen-
tation (r, θ) if r ≥ 0 and 0 ≤ θ < 2π.

9.6. GRAPHS IN POLAR COORDINATES 365

9.6 Graphs in Polar Coordinates

Theorem 9.6.1 A curve in polar coordinates is symmetric about the
(a) x-axis if (r, θ) and (r, −θ) both lie on the curve;
(b) y-axis if (r, θ) and (r, π − θ) both lie on the curve;
(c) origin if (r, θ), (r, θ + π) and (−r, θ) all lie on the curve.

Theorem 9.6.2 Let e be a positive number. Let a fixed point F be called the
focus and a fixed line, not passing through the focus, be called a directrix. If
P is a point in the plane, let P F stand for the distance between P and the
focus F and let P D stand for the distance between P and the directrix. Then
the locus of all points P such that P F = eP D is a conic section representing

(a) an ellipse if 0 < e < 1;

(b) a parabola if e = 1;

(c) a hyperbola if e > 1;

The number e is called the eccentricity of the conic.
In particular an equation of the form

ek
r = 1 ± e cos θ

represents a conic with eccentricity e, a focus at the pole (origin), and a
directrix perpendicular to the polar axis and k units to the right of the pole,
in the case of + sign, and k units to the left of the pole, in the case of −
sign.

Also, an equation of the form

ek
r = 1 ± e sin θ

represents a conic with eccentricity e, a focus at the pole, and a directrix
parallel to the polar axis and k units above the pole, in the case of + sign,
and k units below the pole, in the case of − sign.

366CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

9.7 Areas in Polar Coordinates

Theorem 9.7.1 Let r = f (θ) be a curve in polar coordinates such that f is
continuous and nonnegative for all α ≤ θ ≤ β where α ≤ β ≤ 2π + α. Then
the area A bounded by the curves r = f (θ), θ = α and θ = β is given by

A = β 1 r2dθ = 1 β
α2 2α
(f (θ))2dθ.

Theorem 9.7.2 Let r = f (θ) be a curve in polar coordinates such that f
and f are continuous for α ≤ θ ≤ β, and there is no overlapping, the arc
length L of the curve from θ = α to θ = β is given by

L= β (f (θ))2 + (f (θ))2 dθ
=
α r2 + dr 2
β
dθ dθ
α

9.8 Parametric Equations

Definition 9.8.1 A parametrized curve C in the xy-plane has the form

C = {(x, y) : x = f (t), y = g(t), t ∈ I}

for some interval I, finite or infinite.
The functions f and g are called the coordinate functions and the variable

t is called the parameter.

Theorem 9.8.1 Suppose that x = f (t), y = g(t) are the parametric equa-
tions of a curve C. If f (t) and g (t) both exist and f (t) = 0, then

dy g (t)
=.

dx f (t)
Also, if f (t) and g (t) exist, then

d2y f (g)g (t) − g (t)f (t)
=.

dx2 (f (t))2
At a point P0(f (t0), g(t0)), the equation of

9.8. PARAMETRIC EQUATIONS 367

(a) the tangent line is

y − g(t0) = g (t0) (x − f (t0))
f (t0)

(b) the normal line is

y − g(t0) = −f (t0) (x − f (t0))
g (t0)

provided g (t0) = 0 and f (t0) = 0.

Theorem 9.8.2 Let C = {(x, y) : x = f (t), y = g(t), a ≤ t ≤ b} where f (t)
and g (t) are continuous on [a, b]. Then the arc length L of C is given by

b

L = [(f (t))2 + (g (t))2]1/2dt

a

b dx 2 dy 2 1/2
+ dt.
= dt dt

a

Theorem 9.8.3 Let C = {(x, y) : x = f (t), y = g(t), a ≤ t ≤ b}, where f (t)
and g (t) are continuous on [a, b].

(a) If C lies in the upper half plane or the lower half plane and there is no
overlapping, then the surface area generated by revolving C around the
x-axis is given by

b

2πg(t) (f (t))2 + (g (t))2 dt.

a

(b) If 0 ≤ f (t) on [a, b], (or f (t) ≤ 0 on [a, b]) and there is no overlapping,
then the surface area generated by revolving C around the y-axis is

b

2πf (t) (f (t))2 + (g (t))2 dt.

a

368CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

Definition 9.8.2 Let C = {(x(t), y(t)) : a ≤ t ≤ b} for some interval I.
Suppose that x (t), y (t), x (t) and y (t) are continuous on I.

(a) The arc length s(t) is defined by

t

s(t) = [(x (t))2 + (y (t))2]1/2dt.

a

(b) The angle of inclination, φ, of the tangent line to the curve C is defined

by

φ(t) = arctan y (t) dy .
= arctan
x (t) dx

(c) The curvature κ(t), read kappa of t, is defined by
dφ |x (t)y (t) − y (t)x (t)|
ds = [(x (t))2 + (y (t))2]3/2 .

(d) The radius of curvature, R, is defined by
1

R(t) = .
κ(t)