Practice_Exam - Practice Test For Extraordinary People And...

Practice Test For Extraordinary People And a Few Species of
Swallow And Dolphin; Also “Calculus And You“: A Self-Help
Guide
David Imberti
April 6, 2010
1 A Small Practice Test
The following have their answers worked out below (so, spoiler warning!), consider these as a warm-up:
(1)
x
x 2 +1
dx
(2) cos ( θ ) cos ( sin ( θ )) d θ
(3) sec 4 ( θ )
(4) sin 4 ( θ )
(5)
1
√
x 2 +2 x
dx
(6)
x
3
x 2 - 1
dx
Compute the following integrals (take your time, relax) (I have the solutions on my person, ask me if
you’re unsure):
(1) ln ( x ) dx
(2) arcsin ( x ) dx
(3)
x
4
+1
x 3 + x 2 - x - 1
dx
(4) ( y 2 - y )
- 1
dy
(5) θ tan
2
( θ ) d θ
(6)
1
√
x +1+
√
x
dx
(7) n = 3, approximate f ( x ) =
1
x
between 1 and 2 with the Trapezoidal rule.
(8) Connect the following dots: ( - 2 , 0) , ( - 2 , 2) , ( - 1 , 2) , ( - 1 , 1) , (1 , 1) , (1 , 3) , (3 , 3) , (3 , 0) , ( - 2 , 0). Find the
coordinates of the centroid of the block region inside these points.
(9) Find the surface area by rotating x =
1
3
( y 2 + 2)
3
2
, 1 ≤ y ≤ 2 about the x-axis.
(10) Is
∞
1
ln | x + 1 | - ln | x | dx convergent or divergent?
You might also want to take a look at:
p. 448 25,73 p. 518 21,23,33,18,56,57,2,30,41,46,47 p. 549 34 p. 561 3ab, 12 p. 694 16, 19 p. 759 1-6, 8,
27
Or old exams.
2 Calculus and You
Q: OMIGOSH, THERE’S THIS INTEGRAL AND I DON’T KNOW HOW TO . . . WHAT’S THIS . . . I
DON’T EVEN-
A: O.k., don’t panic, grab your towel.
Remember, you’ve got only a handful of methods to integrate with that you need to remember:
(0) It’s an integral you already know.
1

(1) Direct Substitution.
(2) Trigonometric identities.
(3) Trigonometic substitution.
(4) Partial Fractions.
(5) Integration by Parts.
If you want to go about this methodically, I would go down the list in the order I’ve listed.
Q: COULD YOU EXPLAIN EVERYTHING IN THAT LIST?
A: Sure, what specifically?
2.1 0. Integrals We Know: You’re Half-way There!
Q: WHAT INTEGRALS SHOULD I JUST MEMORIZE?
A: O.k., I’m going to talk a little bit about the more complicated integrals we’ve already done. (you
should know things like cos ( θ ) d θ = sin ( θ ) , e
x
dx = e
x
, x
a
dx =
1
a +1
x
a +1
(for a = - 1, else
x
- 1
dx = - x
- 2
))
Here are the common integrals that have funky forms that you might forget/should memorize:
(Common 1):
1
√
1 - x 2
dx = arcsin ( x ) + C
(Common 2):
1
1+ x 2
dx = arctan ( x ) + C
(Common 3): sec ( θ ) d θ = ln | sec ( θ ) + tan ( θ ) | + C
(Common 4): tan ( θ ) d θ = - ln | cos ( θ ) | + C
Also, these are common trigonometric integrals, but they’re easy to forget when you’re chugging down in
the mood of trigonometric substitution:
(Common 5): sec 2 ( θ ) d θ = tan ( θ ) + C
(Common 6): sec ( θ ) tan ( θ ) d θ = sec ( θ ) + C
2.2 1. Substituting Out For a Better You
Q: HOW DO I KNOW WHAT TO SUBSTITUTE FOR?