Math 2263
Review - Integration, Symmetry, etc.
Name (PRINT):
1. Definitions:
(a) A function f(x) is an even function iff f(−x) = f(x) for all x in the domain of f(x).
(b) A function f(x) is an odd function iff f(−x) = −f(x) for all x in the domain of f(x).
2. Examples:
(a) You should remember and be able to explain why cos(−θ) = cos(θ) for any real number θ. This
shows that cosine is an even function.
(b) You should remember and be able to explain why sin(−θ) = − sin(θ) for any real number θ. This
shows that cosine is an odd function.
p
√
√
(c) Let f(θ) = r 4 − r 2 . Then f(−r) = (−r) 4 − (−r)2 = −r 4 − r 2 = −f(r), so f(r) is an odd
function.
(d) Let f(r) = θ sin θ. Then f(−θ) = (−θ) sin(−θ) = (−θ)[− sin(θ)] == θ sin θ, so f(θ) is an even
function.
3. Check: Draw a graph of any odd continuous function f(x) satisfying f(2) = −3 and f(5) = 1. How about
if f(x) is even, f(−3) = −2 and f(5) = 0 ?
4. Examples: Determine whether the following functions are even, odd, both, or neither. Justify.
(a) f(x) =
x
x2 +1
(b) g(x) = tan x
(c) a(x) =
√
x2
(d) h(x) = ex
2
(e) f(x) = cos x3
(f) g(x) = sin2 x
5. Symmetry: You should understand graphically or be able to use u-substitution to show
(a) If f(x) is an even function continuous on [−b, b], then
Z
b
f(x) dx = 2
Z
b
f(x) dx.
0
−b
(b) If f(x) is an odd function continuous on [−b, b], then
Z
b
f(x) dx = 0.
−b
6. Periodicity: You should understand and be able to easily show that
due to the 2π - periodicity of the sine and cosine functions, in fact
Z b
Z b
cos θ dθ =
sin θ dθ = 0 if [a, b] is any interval of length 2π.
a
Z
0
2π
cos θ dθ =
Z
2π
sin θ dθ = 0. So
0
a
7. Trig Review: Given a real number θ, you should understand the definitions of sin(θ) and cos(θ). Since
the circumference of the unit circle is C = 2πr = 2π, you should understand the values of the trig functions
at the quadrant angles θ = 0, π/2, π and 3π/2. You should understand how to use basic geometry to find
the trig values of sin(θ) and cos(θ) (and hence the other four trig functions) at θ = π/6, π/4 and π/3.
With this, you should finally be able to count through the multiples of θ = π/6, π/4 and π/3 and reason
out the corresponding values of the various trig functions.
8. Evaluate each integral. Show all work.
(a)
Z
2π
Z
π
Z
π
(n)
Z
sin2 θ cos θ dθ
(o)
Z
sin3 θ dθ
(p)
Z
cos3 θ dθ
(q)
Z
tan θ dθ
dθ
(r)
Z
sec θ tan θ dθ
θ cos θ dθ
(s)
Z
tan2 θ dθ
4 − x2 dx
(t)
Z
sec 4 θ dθ
p
4 − x2 dx
(u)
Z
sin2 θ dθ
(v)
Z
cos2 θ dθ
x sin2 x dx
cos θ dθ
0
(b)
cos θ dθ
0
(c)
cos 2θ dθ
0
(d)
Z
Z
π/2
(e)
Z
π/2
Z
2
Z
2
Z
1
1
dx
x2 + 1
Z
1
x dx
x2 + 4
(w)
Z
dx
x 2 + a2
(x)
Z
cos kθ dθ
−π/2
(f)
−π/2
(g)
p
−2
(h)
x
−2
(i)
0
(j)
0
(k)
Z
(l)
Z
e
(m)
Z
ax ebx dx
kx
a
x
0
dx
(y)
Z
(z)
Z
√
p
x
dx
− x2
a2
π/2
−π/2
2
a2 − x2 dx
1
(sin θ + 1) dθ
2
9. Definite Integrals and Symmetry: Suppose that f(x) and g(x) are continuous on (−∞, ∞). Suppose
that f(x) is an odd function and g(x) is an even function. Given
Z
2
f(x) dx = A,
0
Z
1
g(x) dx = B, and
0
Z
2
g(x) dx = C,
−2
compute the following:
(a)
Z
2
Z
2
Z
2
Z
2
Z
1
Z
0
g(x) dx =
0
(b)
f(x) dx =
−2
(c)
g(x) dx =
1
(d)
[f(x) − g(x)] dx =
0
(e)
[3f(x) + 2g(x)] dx =
−1
(f)
[f(x) + g(x)] dx =
−2
10. Symmetry: Here is a Calculus 1 optimization problem: A house is a = 90 yards from the nearest point
P on a straight river bank. A barn is b = 50 yards from the nearest point Q on the river and Q is l = 120
yards downstream from P .
If you start at the house, go to the river to fill up a pail of water, and then to the barn, minimize the
distance traveled. How should you proceed to minimize the distance traveled? What is least distance you
are required to walk?
Begin by thinking about the problem (with pictures!) and various scenarios. In almost any problem
situation it is helpful to imagine yourself as the player and to think about extreme cases: What distances
are required if you were to go directly to P , and then to the barn versus the other extreme, going directly
to Q and then to the barn?
Although not require, a convenient way to model the problem is to place P at the origin and Q on the
positive x-axis. Introduce the appropriate function, minimize using the derivative.
Finally, answer the questions using complete sentences, aided by a diagram if necessary.
Alternatively, can you solve the problem without calculus, instead using the power of symmetry?
Extra Credit: By interpreting as an area, evaluate
Z
0
2
p
16 − x2 dx. Give sufficient explanation.
3