Math 202
Assignment 1: Hyperbolic Fuctions
The due date for this assignment is ..................
Reading assignment: Section 3:11
..................................................................................................................................................
1. Find the numerical value of each expression
(a) sinh (ln 2)
(b) sinh 2
2. Prove the identity
sinh (x + y) = sinh x cosh y + cosh x sinh y
coth2 x
1 = csc h2 x
x2 1
tanh (ln x) = 2
x +1
3. If coshx =
5
and x
3
0; …nd the values of the other hyperbolic functions at x:
4. Find the derivatives. Simplify where possible.
f (t) = sec h2 (et )
y = sinh (cosh x)
1 cosh x
G (x) =
1 + cosh x
p
y = x tanh 1 x + ln 1 x2
p
x
y = x sinh 1
9 + x2
3
1
Math 202
Assignment 2: Indeterminate Forms and L’Hospital’s
Rule
The due date for this assignment is ..................
Reading assignment: Section 4:4
..................................................................................................................................................
1. Find the limit. Use L’hospital’s Rule where appropriate. If L’hospital’s Rule doesn’t
apply, explain why
(a) lim
x!1
(b) lim
xa 1
xb 1
p
1 + 2x
p
1
4x
x
x!0
x
x3
x!0 3x
1
a
x
ax + a
(d) lim
x!1
(x 1)2
(c) lim
1
(e) lim cot (2x) sin (6x)
x!0
(f) lim x3 e
x2
x!1
(g) lim+ ln (x) tan
x!1
(h) lim
x!1
(i) lim (x
x!1
x
x
1
ln x)
(j) lim+ [ln (x7
x!1
x
2
1
ln x
1)
ln (x5
1)]
(k) lim+ (4x + 1)cot x
x!0
(l) lim+ (cos x)1=x
2
x!0
2
Math 202
Assignment 3: Antiderivatives
The due date for this assignment is ..................
Reading assignment: Section 4:9
..................................................................................................................................................
1. Find the most general antiderivatives of the function (check your answer by di¤erentiation)
(a) f (x) = (x + 1) (2x
p
(b) f (x) = 2
(c) f (x) =
1)
1 + t + t2
p
t
2. Find f
f 000 (t) = cos t
4
f 0 (t) =
; f (1) = 0
1 + t2
f 0 (t) = 2 cos t + sec2 t;
<t< ; f
2
2
0
1=3
f (x) = x
; f (1) = 1; f ( 1) = 1
3
=4
f 00 (x) = x 2 ; x > 0; f (1) = 0; f (2) = 0
3. A particle is moving with the given data. Find the position of the particle
v (t) = sin t
cos t;
s (0) = 0
a (t) = 2t + 1; s (0) = 3; v (0) =
2
3
Math 202
Assignment 4: The De…nite Integral
The due date for this assignment is ..................
Reading assignment: Section 5:2
..................................................................................................................................................
1. Evaluate the integral by interpreting it in terms of areas
Z 2
(a)
(1 x) dx
1
2
Z
(b)
1
jxj dx
2. Write as a single integral in the form
Z
b
f (x) dx :
a
Z
3. If
Z
5
f (x) dx = 12 and
f (x) dx +
2
Z
Z
Z
Z
5
f (x) dx
2
5
f (x) dx = 3:6; …nd
4
1
4. Find
2
Z
1
f (x) dx
2
4
f (x) dx
1
5
f (x) dx if
0
f (x) =
3
x
for x < 3
for x 3
5. Use property 8 to estimate the value of the integral
Z 4
p
x dx
1
Z
=3
tan x dx
=4
4
Math 202
Assignment 5: The Fundamental Theorem of Calculus
The due date for this assignment is ..................
Reading assignment: Section 5:3
..................................................................................................................................................
1. Use Part 1 of the Fundamental Theorem of Calculus to …nd the derivative of the
function
Z ex
(a) h (x) =
ln (t) dt
1
Z 1
u3
(b) y =
du
2
1 3x 1 + u
2. Evaluate the integral.
Z 2
(a)
(1 + 2y)2 dy
1
Z 1
(b)
(xe + ex ) dx
0
Z 1
eu+1 du
(c)
Z 1
(d)
f (x) dx where
0
f (x) =
sin x
cos x
if 0 x < n2
if n2 x
3. What is wrong with the equation?
Z 1
x 3 1
3
x 4 dx =
] 2=
3
8
2
4. Find the derivative of the function
Z sin x
y=
ln (1 + 2v) dv
cos x
5
Math 202
Assignment 6: Inde…nite Integrals and The Net
Change
The due date for this assignment is ..................
Reading assignment: Section 5:4
..................................................................................................................................................
1. Find the general inde…nite integral.
Z
(a)
(1 + tan2 ) d
2. Evaluate the integral.
(a)
Z
1
(x10 + 10x ) dx
0
(b)
(c)
Z
0
Z p
0
(d)
(e)
n4
Z
Z
1 + cos2
cos2
3n2
p
p
1n 3 2
t
t4
0
2
1
jx
d
dr
1 r2
1
dt
1
2 j x j j dx
6
Math 202
Assignment 7: The Substitution Rule
The due date for this assignment is ..................
Reading assignment: Section 5:5
..................................................................................................................................................
1. Evaluate the inde…nite integral.
Z
(a)
5t sin (5t ) dt
Z
cos
d
(b)
sin2
Z
sin 2x
(c)
dx
1 + cos2 x
Z
sin x
(d)
dx
1 + cos2 x
Z
dx
p
(e)
1 x2 sin 1 x
Z
x
(f)
dx
1 + x4
Z
1+x
(g)
dx
1 + x2
2. Evaluate the de…nite integral.
(a)
(b)
(c)
Z
Z
n4
2
1
Z 1
0
(x3 + x4 tan x) dx
n4
p
x x
1 dx
dx
p 4
(1 + x)
7
Math 202
Assignment 8: Integration by Parts
The due date for this assignment is ..................
Reading assignment: Section 7:1
..................................................................................................................................................
1. Evaluate the integral.
Z
(a)
(x2 + 2x) cos x dx
Z
p
(b)
ln ( 3 x) dx
Z
(c)
x 2x dx
(d)
(e)
(f)
(g)
Z
Z
Z0
Z
x e2x
dx
(1 + 2x)2
1
y
dy
e2y
cos x ln (sin x) dx
2
x4 (ln x)2 dx
1
2. First make a substitution and then use intgration by parts to evaluate the integral.
Z
p
(a)
cos x dx
(b)
(c)
Z
Z
p
p
3
cos
2
d
n2
x ln (1 + x) dx
8
Math 202
Assignment 9: Trigonometric Integrals
The due date for this assignment is ..................
Reading assignment: Section 7:2
..................................................................................................................................................
1. Evaluate the integral.
Z
(a)
cos4 (2x) dx
0
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Z
n2
sin2 x cos2 x dx
Z0
t sin2 t dt
Z
cos2 x tan3 x dx
Z
Z
Z
Z
Z
cos5 x
p
dx
sin x
tan5 x dx
x sec x tan x dx
n2
cot5
csc3
d
n4
csc x dx
Z
n6 p
1 + cos 2x dx
0
(k)
Z
x tan2 x dx
9
Math 202
Assignment 10: Trigonometric Substitution
The due date for this assignment is ..................
Reading assignment: Section 7:3
..................................................................................................................................................
1. Evaluate the integral.
Z p 2
x
9
dx
(a)
x3
Z p
1 + x2
(b)
dx
x
Z 0:6
x2
p
dx
(c)
9 25x2
0
Z
x
p
(d)
dx
x2 + x + 1
Z
p
(e)
x2 + 2x dx
Z
p
(f)
x 1 x4 dx
10
Math 202
Assignment 11: Integration of Rational Function by
Partial Fractions
The due date for this assignment is ..................
Reading assignment: Section 7:4
..................................................................................................................................................
1. Evaluate the integral.
Z
4x
(a)
dx
3
2
x +x +x+1
Z 3
x + x2 + 2x + 1
(b)
dx
(x2 + 1) (x2 + 2)
Z
x+4
(c)
dx
2
x + 2x + 5
Z
1
(d)
dx
x3 1
2. Make a substitution to express the integrand as a rational function and then evaluate
the integral.
Z p
x+1
dx
(a)
x
Z
dx
p
(b)
2
x +x x
Z
x3
p
dx
(c)
3
2+1
x
Z
p
1
p
p
(d)
dx [Hint: Substitute u = 6 x]
3
x
x
Z
2x
e
(e)
dx
2x
e + 3ex + 2
Z
sec2 x
(f)
dx
tan2 x + 3 tan x + 2
Z
dx
(g)
1 + ex
11
Math 202
Assignment 12: Strategy for Integration
The due date for this assignment is ..................
Reading assignment: Section 7:5
..................................................................................................................................................
1. Evaluate the integral.
Z
x
(a)
ex + e dx
Z
1
dx
(b)
1 + ex
Z
sec x tan x
(c)
dx
sec2 x sec x
Z
(d)
tan2 d
Z p
x
(e)
dx
1 + x3
Z
3
(f)
x5 e x dx
Z
(g)
x3 (x 1) 4 dx
Z
(h)
cos x cos3 (sin x) dx
Z
sin 2x
dx
(i)
1 + cos4 x
Z
1
p
(j)
p dx
x+1+ x
Z p3 p
1 + x2
(k)
dx
x2
Z1
xex
p
dx
(l)
1 + ex
Z
(m)
x sin2 x cos x dx
Z
p
(n)
1 sin x dx
12
Math 202
Assignment 13: Improper Integral
The due date for this assignment is ..................
Reading assignment: Section 7:8
..................................................................................................................................................
1. Determine weather each integral is convergent or divergent. Evaluate those that are
convergent.
Z 1
(a)
sin2 d
0
(b)
(c)
Z
Z
1
1
3
x2
0
(d)
(e)
Z
Z
x2
dx
9 + x6
0
1
2
dx
6x + 5
e1nx
dx
x3
x2 ln x dx
0
2. Sketch the region and …nd its area (if the area is …nite)
S = f(x; y) j x
1; 0
y
e xg
3. Use the Comparison Theorem to determine weather the integral convergent or divergent.
Z 1
x
dx
(a)
3
x +1
0
Z 1
2+e x
dx
(b)
x
1
Z 1
x+1
p
(c)
dx
x4 x
1
13
(d)
(e)
(f)
Z
Z
1
0
1
arctan x
dx
2 + ex
0
sec2 x
p dx
x x
0
sin2 x
p dx
x
Z
4. The integral
Z
0
1
p
1
dx
x (1 + x)
is improper for two reasons: The interval [0; 1) is in…nite and the integrand has an
in…nite discontinuity at 0. Evaluate it by expressing it as a sum of improper integrals
of Type 2 and Type 1 as follows:
Z 1
Z 1
Z 1
1
1
1
p
p
p
dx =
dx +
dx
x (1 + x)
x (1 + x)
x (1 + x)
0
0
1
14
Math 202
Assignment 14: Areas Between Curves
The due date for this assignment is ..................
Reading assignment: Section 6:1
..................................................................................................................................................
1. Sketch the region enclosed by the given curves. Decide weather to integrate with
respect to x or y. Draw a typical approximating rectangle and label its height and
width. Then …nd the area of the region.
(a) y =
1
;
x
y=
1
;
x2
x=2
2. Sketch the region enclosed by the given curves and …nd its area.
(a) y = ex ;
(b) y = tan x;
y = xex ;
x = 0:
y = 2 sin x;
(c) y = cos x;
y=1
1
(d) y = ;
y = x;
x
n3
x
cos x;
0 x
1
y = x; x > 0
4
n3
3. Use calculus to …nd the area of the triangle with the given vertices.
(a) (0; 0) ;
(3; 1) ;
(1; 2)
15
Math 202
Assignment 15: Volumes
The due date for this assignment is ..................
Reading assignment: Section 6:2
..................................................................................................................................................
1. Find the volume of the solid obtained by rotating the region bounded by the given
curves about the speci…ed line. Sketch the region, the solid, and a typical disk or
washer.
(a) y = ln x; y = 1;
y = 2;
x = 0;
about the y
1
(b) y = x2 ;
y = 5 x2 ;
about the x axis
4
1
(c) y = x2 ;
x = 2;
y = 0;
about the y axis
4
(d) y = 1 + sec x;
y = 3;
about y = 1
(e) y = sin x;
(f) x = y 2 ;
(g) y = x;
y = cos x; 0
x = 1;
p
y = x;
x
about x = 1
n4;
about x = 2
16
about y =
axis
1
Math 202
Assignment 16: Arc Length
The due date for this assignment is ..................
Reading assignment: Section 8:1
..................................................................................................................................................
1. Find the exact length of the curve
1p
y (y 3) ; 1 y 9
3
(b) y = ln (cos x) ;
0 x
n3
1
(c) y = 3 + cosh 2x;
0 x 1
2
1
1
(d) y = x2
ln x;
1 x 2
4
2
1
(e) y = ln (1 x2 ) ;
0 x
2
(a) x =
2. Find the arc length function for the curve y = 2x3n2 with starting point P0 (1; 2) :
17
Math 202
Assignment 17: Area of Surface of Revolution
The due date for this assignment is ..................
Reading assignment: Section 8:2
..................................................................................................................................................
1. Find the exact area of the surface obtained by rotating the curve about the x
(a) 9x = y 2 + 18; 2
(b) y = sin x;
0
3
(c) y =
x
1
+ ;
6
2x
x
x
6
1
1
2
x
1
2. The given curve is rotated about the y
(a) y =
p
3
x;
1
(b) y = x2
4
1 y
1
ln x;
2
axis
axis. Find the area of the resulting surface.
2
1
x
2
18