MATH 113, CHAPTER 5, INTEGRATION
1. Estimating with finite sums
Example 1. Approximating Area above the x-axis, below the graph of y = 1−x2 , and between the vertical
line x = 0 and x = 1?
Note: upper sum, lower sum, midpoint rule, distance traveled, average of a nonnegative function.
Figure 1.
Example 2. Estimate the height of a projectile
The velocity function of a projectile fired straight into the air is f (t) = 160 − 9.8tm/sec. Use the
summation techniques to estimate how far the projectile rises during the first 3 sec.
Example 3. The average of of sin x
Estimate the average of the function f (x) = sin x on the interval [0, π]. (see figure 2.)
Note 4. Estimating with finite sums.
First we subdivide the interval into subintervals, treating the appropriate function f as if it were
constant over each particular subinterval. Then we multiply the width of each subinterval by the value of
f at some point within it, and add these products together.
If the interval [a, b] is subdivided into n subintervals of equal widths ∆x = (b − a)/n, and if f (ck ) is
the value of f at the chosen point ck in the k−th subinterval, this process gives a finite sum of the form
f (c1 )∆x + f (c2 )∆x + f (c3 )∆x + · · · + f (cn )∆x.
The choices of the ck could maximize or minimize the value of f in the kth subinterval, or give some
value in between.
1
2
MATH 113, CHAPTER 5, INTEGRATION
Figure 2.
2. Sigma Notation and Limits of Finite Sums
n
X
Notation 5. Sigma notation, see page 336 for the summation symbol,
ak
k=1
The index k ends at k = n, the index k starts at k = 1, ak is a formula for the k-th term.
Example 6. Using Sigma Notation
5
X
1. 1 + 2 + 3 + 4 + 5 =
i
i=1
2. 1 + 3 + 5 + · · · + 999 =
500
X
(2i − 1)
i=1
1000
X
3. 12 + 22 + 32 + · · · + 10002 =
2
2
2
2
4. 1 + 2 + 3 + · · · + n =
n
X
(i2 )
i=1
(i2 )
i=1
Note 7. Algebra Rules for Finite Sums
n
n
n
X
X
X
1. Sum Rule:
(ak ) +
(bk ) =
(ak + bk )
k=1
2. Difference Rule:
n
X
k=1
(ak ) −
k=1
n
X
k=1
(bk ) =
k=1
n
X
(ak − bk )
k=1
3. Constant Multiple Rule: (Any number c)
4. Constant Value Rule:
n
X
n
X
c · ak = c ·
k=1
c=c·n
k=1
Theorem 8. Some Summations.
1. The Sum of the First n Integers:
2. The first n squares:
n
X
k=1
3. The first n cubes:
n
X
k=1
Example 9. Evaluate
n
X
a.
(2 + 3i)
i=1
n
X
k=
k=1
n(n + 1)
2
n(n + 1)(2n + 1)
k2 =
6
k3 = (
n(n + 1) 2
)
2
n
X
k=1
ak
MATH 113, CHAPTER 5, INTEGRATION
b.
n
X
3
i(i − 1)
i=1
c. 1 + 3 + 5 + · · · + 999
n
X
i
1
d. limn→∞
( )2 ·
n
n
i=1
n
X
i−1 2 1
) ·
e. limn→∞
(
n
n
i=1
n
X
2i − 1 2 1
f. limn→∞
(
) ·
n
n
i=1
Example 10. Graph the function f (x) = x2 + 1 over the interval [0, 3].
Partition the interval into four subintervals of equal length. then add to your sketch the rectangles
4
X
associated with the Riemann sum
f (ck )∆xk . given that ck is (a) left-hand endpoint, (b) right-hand
k=1
endpoint, (c) midpoint of the kth subinterval.
Example 11. Use limits of upper sums to calculate the area of the region of y = x2 + 1, [0, 3].
Note 12. The limit of Finite approximation to an Area.
Riemann Sums and Area
see detailed textbook.
partition → approximate → sum → limit
3. The Definite Integral(text 343-352)
Z b
Notation and Existence of the Definite Integral
f (x)dx.
a
Upper limit of the integration, lower limit of integration, the function is the integrand, x is the variable
of integration.
Figure 3.
see page 344 for notation
Example 13. Find the average value of f (x) =
√
4 − x2 on [−2, 2].
4
MATH 113, CHAPTER 5, INTEGRATION
Figure 4.
Z b
Example 14.
cdx = c(b − a)
a
Z b
1
xdx = (b2 − a2 )
2
Zab
1
x2 dx = (b3 − a3 )
3
a
Z
1
Example 15. Use the Max-Min Inequality to find upper and lower bounds for the value of
0
1
dx.
1 + x2
4. The Fundamental Theorem of Calculus
Z x
If f (t) is a continuous function on [a, b], then the integral
f (t)dt defines a new function F (x).
Z
Example 16. Let f (t) = t, a = 0 Find
case 1. x > 0
case 2. x < 0
a
x
f (t)dt
a
Theorem 17. The Fundamental Theorem of Calculus Part
Z x 1
Let f (t) be a continuous function on [a, b], if F (x) =
f (t)dt , then g(x) is continuous on [a, b], and
a
F 0 (x) = f (x) for all x ∈ (a, b). In other words, F (x) is an antiderivative of f (x).
0
Example 18.
Z xFind g (x)
1
√
a. g(x) =
dt
1
+ t2
Z 01
b. g(x) =
sin(t)dt
Zxx2
c. g(x) =
t sin(t)dt
Z2 1
t2 dt
d. g(x) =
sin
x
Z x
1
e. g(x) =
dt
2
x2 1 + t
Theorem 19. The Fundamental Theorem of Calculus Part 2
Z
F (b) − F (a) =
F (x)|ba
Example
Z 1 20. Evaluate
a.
x2 dx
Z −2
4
3√
4
(
b.
x − 2 )dx
2
x
Z1 2 3
x +1
dx
c.
x2
1
b
f (x)dx =
Let f (t) be a continuous function on [a, b], if F (x) is an antiderivative of f (x), then
a
MATH 113, CHAPTER 5, INTEGRATION
5
Figure 5.
Z
1
d.
(t − 2)(t + 1)dt
Z 00
e.
sin tdt
π
6
Example 21. Find the area bounded by the x-axis, y = x2 − 1, x = −2 and x = 3.
Z x+1
9
Example 22. Find the linearization of f (x) = 2 −
dt.
1
+
t
2
5. Indefinite Integrals and the Substitution Rule, and Area between Curves
Theorem 23. If u = g(x) is a differentiable function whose range is an interval I and f is continuous
on ZI, then
Z
f (g(x))g 0 (x)dx = f (u)du
How to evaluate
Z
0
1. Substitute u = g(x), then du = g (x)dx, we obtain f (u)du
2. Integrate w.r.t. u.
3. Replace u by g(x).
Example
24. Evaluate
Z
2x
√
a.
dx
1 + x2
Z
b.
Z
x2 (x3 + 1)4 dx
sin4 x cos xdx
√
Z
(1 + 3 x)2
√
dx
d.
3
x2
c.
Example 25. Solve the initial value problem.
ds
= 12t(3t2 − 1)3 , s(1) = 3.
dt
6
MATH 113, CHAPTER 5, INTEGRATION
Theorem 26. If g 0 is a continuous function on [a, b] and f is continuous on the range of g, then
Z b
Z g(b)
f (g(x))g 0 (x)dx =
f (u)du
a
g(a)
Example
Z 1 27. Evaluate
t
dx
a.
(1 + t2 )3
Z 0 π6
b.
cos 2xdx
Z0 π4
c.
cos3 xdx
0
Theorem 28. Z
If f is a continuous
Z a function on [−a, a]
a
if f is even,
f (x)dx = 2
f (x)dx
0
Z a−a
if f is odd,
f (x)dx = 0.
−a
Example
Z 20 29. Evaluate
4
a.
x5 (x2 + 1) 5 dx
−20
Z 1
b.
x4 dx
−1
Z π2
cos xdx
c.
−π
2
Definition 30. Area between curves If f and g are continuous with f (x) ≥ g(x) throughout [a, b], then
the area of the region between the curves y = f (x) and y = g(x) from a to b is the integral of (f − g)
from a to b:
Z b
A=
[f (x) − g(x)]dx
a
Example 31. find the areas of the regions by the lines and curves.
1. x = y 2 and x = y + 2.
2. x = y 2 and x = y 3 .
3. y = x, y = 1 and y = x2 /4