CALCULUS 2 - Grant's Tutoring · Grant’s Tutoring (text or call (204) 489-2884) DO NOT RECOPY . Grant’s Tutoring is a private tutoring organization and is in no way affiliated
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Homework for Lesson 1: Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 189 #1; p. 190 #2; p.191, #3;
p. 192 #2; p. 193 #3; p. 194 #2 (b); p. 195 #9; p. 196 #5; p. 197 #5; p. 198 #2; p. 199 #2; p. 202 #1 (a); p. 203 #4 (a); p. 211 #1; p. 213 #2; p. 224 #5 (a) and (b). Note: the solutions to these questions begin on page 229.
Lesson 2: The Fundamental Theorem of Calculus
( ) ( )u
af t dt f u u⌠
⌡
′= ⋅ ′
Average Value of f(x) from a to b: ( )1 b
a
f x dxb a− ∫
Be sure to memorize all the Elementary Integrals on page 1.
1. Solve the following definite and indefinite integrals:
Homework for Lesson 2: While doing this homework make index cards for question 1 only (see page 5 for an
explanation). Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 189 #2; p.191, #1; p. 193 #2;
p. 194 #2 (a); p. 195 #6; p. 196 #2 and #3; p. 198 #1 (e) and #3; p. 199 #1 (e) and #3; p. 200 #2 and #5; p. 201 #3; p. 202 #1 (b); p. 203 #4 (b) and (c); p. 205 #1 (a); p. 209 #2; p. 214 #9 Part B; p. 216 #9; p. 217 #3. Note: the solutions to these questions begin on page 229.
Lesson 3: Riemann Sums The Definition of the Definite Integral:
( ) ( ) ( )0
1 1
lim * limb n n
iP ni ia
b a ib af x dx f x x f a
n n→ →∞= =
− −= ∆ = ⋅ +
∑ ∑∫
Summation Formulas to Memorize:
1
1n
i
n=
=∑ ( )1
12
n
i
n ni
=
+=∑
( )( )2
1
1 2 16
n
i
n n ni
=
+ +=∑
1. Compute the left and right Riemann sums for the following functions over the given
interval using 4 equal partitions. Include a sketch of the region. (a) ( ) 23 4f x x= − on [0, 4] (b) ( ) 2 4 3f x x x= − + on [–1, 7]
2. Compute the lower and upper Riemann sums for the following functions over the given
interval using 4 equal partitions. Include a sketch of the region. (a) ( ) 23 4f x x= − on [0, 4] (b) ( ) 2 4 3f x x x= − + on [–1, 7]
3. Compute the Riemann sum for the following functions over the given interval using n
equal partitions. Then determine the limit as n approaches infinity (n →∞ ). Check your answer by computing the associated definite integral.
(a) ( ) 23 4f x x= − on [0, 4] (b) ( ) 2 4 3f x x x= − + on [–1, 7]
Homework for Lesson 4: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #3 and #6; p. 190 #1 (a)
and (d); p.191, #2 (a); p. 192 #1 (a) and (b) and #3; p. 193 #1 (a) and (b); p. 194 #1 (a); p. 195 #1; p. 196 #1 (b); p. 197 #2; p. 198 #1 (a) and (f); p. 199 #1 (a) and (b); p. 200 #3 (a); p. 201 #2 (a) and (b); p. 202 #2 (a); p. 203 #1 (a) and #2 (a); p. 205 #1 (b) and (c); p. 207 #1 (b); p. 209 #3 (a); p. 211 #2 (b); p. 213 #3 (a) and (b); p. 214 #9 Part A; p. 215 #3 (a), (b) and (d); p. 221 #1 (a); p. 223 #1 (a) and (c); p. 225 #1 (a); p. 227 #1 (a) and (c). Note: the solutions to these questions begin on page 229.
Lesson 5: Area between 2 Curves
( )1 2
x b
x a
y y dx=
=
−∫ or ( )1 2
y b
y a
x x dy=
=
−∫
1. Compute the areas of the following bounded regions: (a) 3y x= and 1 3y x= (b) 2x y= and 22 2x y y= − −
(c) 1
yx
= and 2 2 5x y+ = (d) 212
y x= and 2
11
yx
=+
(e) 5xy e= , x = 0, x = 2, and the x-axis (f) 2 consecutive intersections of siny x= and cosy x= 2. Let R be the region inside the circle centred at the origin of radius 3 and above the line
2 3y x= − . Set up BUT DO NOT EVALUATE the integral (or integrals) that would find the area of R.
Homework for Lesson 5: Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 190 #4 (a) and (b) part (i);
p.191, #4; p. 192 #4 (a) and (b); p. 193 #4 (a) and (b); p. 195 #5; p. 196 #4; p. 198 #4; p. 199 #4; p. 200 #4; p. 201 #4; p. 202 #5; p. 203 #3; p. 209 #1; p. 211 #5 (a); p. 213 #1 (a) and (b); p. 215 #1 (a) and (b); p. 217 #2; p. 221 #4 (a); p. 228 #6 (a). Note: the solutions to these questions begin on page 229.
1. Compute the volumes of the following bounded regions, if the region is rotated about: (i) the x-axis (ii) the y-axis (a) 3y x= and 2y x= (b) y x= and 24x y y= − (c) sin 2y x= + and 2y = , 0 x π≤ ≤ (d) 2xy e= , the x-axis, x = 1 and x = 2 2. Given the region bounded by y x= and 2y x= , find the volume if this region is rotated
about the x-axis by using: (a) the “disk” or “washer” method (b) the cylindrical shell method. 3. For the region R bounded by 24y x x= − and 28 2y x x= − , set up integrals for the
following: (a) The volume of the solid created by rotating R about the x-axis. (b) The volume of the solid created by rotating R about the y-axis. (c) The volume of the solid created by rotating R about the line x = –2. (d) The volume of the solid created by rotating R about the line y = 10. (e) The volume of the solid created by rotating R about the line x = 7. 4. Derive the formula for the volume of a right circular cone of height h and base radius r
by setting up and solving a volume integral. 5. Derive the formula for the volume of a sphere of radius r by solving the appropriate
volume integral. 6. A solid has a circular base of radius 3 units. Find the volume of the solid if parallel cross-
Homework for Lesson 6: Repeat questions 1 to 6 in this lesson. Do the following questions from the old exams: p. 190 #3 and #4 (skip part (i));
p.191, #5; p. 192 #4 (c) and #5; p. 193 #4 (skip (b)); p. 194 #4 and #5; p. 195 #7 and #8; p. 196 #6; p. 197 #3; p. 206 #5 (c); p. 207 #4 (c) and (d); p. 210 #6 (c); p. 211 #5 (b); p. 213 #1 (c); p. 215 #1 (c); p. 217 #1; p. 219 #3 and #4; p. 221 #4 (c); p. 224 #6 and #7; p. 226 #5 (a) and (b) and #6; p. 227 #5; p. 228 #6 (b) and (c). Note: the solutions to these questions begin on page 229.
sin2 2sin cosθ θ θ= 1. Evaluate the following: (a) 2cos 2x dx∫ (b) 2 2sin cosx x dx∫
(c) 3sin 2x dx∫ (d) 3 6sin cosx x dx∫
(e) 2 32sin cosx x dx∫ (f) 4tan secx x dx∫
(g) 2
2
1 cotcos
xdx
x
⌠⌡
− (h) ( )21 cot x dx+∫
Homework for Lesson 7: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #4; p. 190 #1 (b); p.191,
#2 (c); p. 192 #1 (d); p. 193 #1 (d); p. 194 #1 (b); p. 195 #2; p. 196 #1 (a); p. 197 #4 (b); p. 198 #1 (b); p. 200 #3 (b); p. 201 #2 (e); p. 202 #3 (a); p. 203 #2 (b); p. 207 #1 (c); p. 211 #2 (a); p. 217 #4 (b); p. 219 #1 (d); p. 225 #1 (c). Note: the solutions to these questions begin on page 229.
1. Evaluate the following: (a) 2 lnx x dx∫ (b) ( )2
ln x dx∫
(c) 1sin 2x dx−∫ (d) 21
0
xx e dx∫
(e) ( )2 sin 4x x dx∫ (f) 2secx x dx∫
(g) 2 cos3xe x dx∫ (h) ( )cos ln x dx∫
Homework for Lesson 8: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #7; p. 190 #1 (c); p.191,
#2 (b); p. 192 #1 (c); p. 193 #1 (c); p. 194 #1 (d); p. 195 #3; p. 196 #1 (d); p. 197 #4 (a); p. 198 #1 (d); p. 199 #1 (d); p. 200 #3 (c); p. 201 #2 (c); p. 202 #3 (b); p. 203 #1 (b); p. 205 #2 (a); p. 207 #1 (d); p. 209 #3 (c); p. 213 #3 (c) and (d); p. 215 #3 (c); p. 217 #4 (a); p. 219 #1 (b); p. 221 #1 (e); p. 223 #1 (b); p. 225 #1 (b); p. 227 #1 (b). Note: the solutions to these questions begin on page 229.
Lesson 9: Integrating by Trig Substitution 1. Evaluate the following:
Homework for Lesson 9: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #5; p.191, #2 (d); p. 193
#1 (e); p. 194 #1 (c); p. 196 #1 (c); p. 198 #1 (c); p. 199 #1 (c); p. 200 #3 (d); p. 201 #2 (d); p. 202 #2 (b); p. 203 #1 (c); p. 205 #2 (c); p. 211 #2 (c); p. 213 #3 (e); p. 215 #3 (e); p. 217 #5; p. 219 #1 (c); p. 221 #1 (b); p. 223 #1 (d); p. 227 #1 (d). Note: the solutions to these questions begin on page 229.
Lesson 10: Integrating Rational Functions 1. Evaluate the following:
(a) 11
xdx
x⌠⌡
−+
(b) 2 4 5
2x x
dxx
⌠⌡
+ +−
(c) 2
26 7
xdx
x x
⌠⌡
+− −
(d) ( )( )( )
2 3 41 2 3x x
dxx x x
⌠⌡
+ +− − −
(e) 4 1x
dxx
⌠⌡ −
(f) ( )( )21 4
dxx x
⌠⌡ − +
(g) ( ) ( )2 2
2 3
dx
x x
⌠⌡ − −
(h) 3 2
dxx x
⌠⌡ −
Homework for Lesson 10: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #8; p. 202 #2 (c); p. 205
#2 (b); p. 207 #1 (a); p. 209 #3 (b); p. 211 #2 (d); p. 213 #3 (f); p. 215 #3 (f); p. 217 #4 (c); p. 219 #2; p. 221 #1 (c); p. 223 #2; p. 225 #2; p. 227 #2. Note: the solutions to these questions begin on page 229.
Lesson 11: L’Hôpital’s Rule 1. Solve the following limits:
(a) 30
sinlimx
x xx→
− (b)
2
10lim
sinx
xx−→
(c)
( )( )
2lim sec cos3
xx x
π−→
(d) 0
lim lnx
x x+→
(e) 23lim
x
xx e→∞
− (f) ( )0
1limx
xxx e→
+
(g) 1
lim cosx
x
x→∞
(h) ( )0
cotlim 1 sin4x
xx+→
+
Homework for Lesson 11: Repeat question 1 in this lesson. Do the following questions from the old exams: p. 205 #3; p. 207 #2; p. 209 #4;
p. 211 #3; p. 213 #4; p. 215 #4; p. 217 #6; p. 220 #6; p. 221 #2; p. 223 #3; p. 225 #3; p. 227 #4. Note: the solutions to these questions begin on page 229.
Lesson 12: Improper Integrals and the Comparison Theorem 1. Determine if the following improper integrals converge or diverge. Evaluate those that
2. Use the Comparison Test to determine if the following integrals converge or diverge.
(a) 3
4
0
dxx x
⌠⌡ +
(b) 3
1
dxx x
⌠⌡
∞
+
(c) 2
2
1
sin xdx
x
⌠⌡
∞
(d)
1
1 xdx
x
⌠⌡
∞
+
(e) 1
2xdx
x e
⌠⌡
∞
+ (f)
1
0
xedx
x
⌠⌡
−
Homework for Lesson 12: Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 205 #4; p. 207 #3; p. 209 #5;
p. 211 #4; p. 214 #5; p. 216 #5; p. 218 #7; p. 220 #7; p. 221 #3; p. 223 #4; p. 225 #4; p. 227 #3. Note: the solutions to these questions begin on page 229.
Lesson 13: Arc Length and Surface Area
2
1dy
dL dxdx
= +
or 2
1dx
dL dydy
= +
If the curve is rotated about the x-axis: 2b
a
S y dLπ= ∫
If the curve is rotated about the y-axis: 2b
a
S x dLπ= ∫
1. Find the lengths of the following curves: (a) 2, 0 1y ex x= + ≤ ≤ (b) ( )ln sin , 6 3x y yπ π= ≤ ≤
2. Set up but do not solve the integrals to find the surface area if the curve 3 2 4y x x= + + on [0, 2] is revolved about:
(a) the x-axis (b) the y-axis (c) the x = 3 line (d) the y = –2 line 3. Find the surface area obtained by rotating the first quadrant region of the curve
21 , 1 5x y x= + ≤ ≤ about: (a) the x-axis (solve this integral) (b) the y-axis (set up the integral, but do not solve it) Homework for Lesson 13: Repeat questions 1 to 3 in this lesson. Do the following questions from the old exams: p. 206 #5 (a) and (b); p. 207 #4 (b)
and (e); p. 210 #6 (a) and (b); p. 211 #5 (c) and (d); p. 214 #6; p. 216 #6; p. 218 #8; p. 220 #5; p. 221 #4 (b); p. 224 #8; p. 226 #5 (c); p. 228 #7. Note: the solutions to these questions begin on page 229.
Lesson 14: Parametric Equations
2 2dx dydL dt
dt dt = +
( )1 2
x b
x a
Area y y dx=
=
= −∫ or ( )1 2
y b
y a
Area x x dy=
=
= −∫
1. For the parametric equations given below find dydx
and 2
2
d ydx
.
(a) 3cos , 5tanx t y t= = (b) 3 24 5, 4 6 9x t y t t= + = − + 2. Find the equation of the line tangent to the parametric curve lnx t= , ty te= at t = 1. 3. Find the equation of the line tangent to the parametric curve 2 3x t= + , 2 2y t t= + at
the point (5, 3). 4. Find the area bounded by the curve 1x t t= − , 1y t t= + and the line y = 5/2.
5. For the curve defined parametrically by ( )2 3x t t= − , ( )23 3y t= − , do the following:
(a) Show the point (0, 0) has 2 tangent lines. (b) Find the points where the curve has either a horizontal or a vertical tangent line. (c) Sketch the curve. (d) Set up but do not solve the integrals expressing: (i) The circumference of the enclosed region. (ii) The area of the enclosed region. (iii) The surface area of the solid created by rotating the enclosed region about
the x-axis. 6. Sketch the parametric curves defined below, by first doing the following: (i) Find the points which have vertical or horizontal tangent lines. (ii) Use the first derivative to establish the directional information of the curve. (iii) Use the second derivative to establish the concavity of the curve. (a) 2 22 , 4x t t y t t= − = − (b) 2 32 , 12x t y t t= = − [Hint: Verify that this curve has two tangents at (24, 0).] Homework for Lesson 14: Repeat questions 1 to 6 in this lesson. Do the following questions from the old exams: p. 206 #6; p. 208 #5; p. 210 #7;
p. 212 #6; p. 214 #7; p. 216 #7; p. 218 #9; p. 220 #8; p. 222 #5; p. 224 #9; p. 226 #7; p. 228 #8. Note: the solutions to these questions begin on page 229.
1. Sketch the following polar curves: (a) r = 5 (b) 2sinr θ= (c) 4 cosr θ= − (d) ( )3 1 cosr θ= −
(e) 1 2sinr θ= + (f) sin2r θ= (g) 4 cos3r θ= (h) , 1r θ θ= ≥ 2. Find the area of one petal of the rose cos2r θ= . 3. Set up but do not solve the integral for the circumference of the cardioid 1 sinr θ= + .
4. Find the slope of the tangent line to 1 cosr θ= + at 6π
θ = .
Homework for Lesson 15: Repeat questions 1 to 4 in this lesson. Do the following questions from the old exams: p. 210 #8; p. 214 #8; p. 216 #8;
p. 218 #10; p. 220 #9; p. 222 #6; p. 226 #8; p. 228 #9. Note: the solutions to these questions begin on page 229.