Drill: Evaluate each sum
• Recall that
n
a
i 1
i
• 12 + 22 + 32 + 42 + 52 =55
 a1  a2  a3  .....  an
5
2
n

n 1
• [3(0) -2] + [3(1) -2] +
[3(2) -2] + [3(3) -2] +
[3(4) -2] = 20
4
 (3k  2)
k 0
4
2
100
(
j

1
)

j 0
• 100 (0 + 1)2 + 100 (1 + 1)2 + 100 (2
+ 1)2 + (3 + 1)2 + 100 (4 + 1)2
=5500
Write the sum in sigma notation
• 1 + 2 + 3 + …..+ 98 +
99
• 0 + 2 + 4 + ….48 + 50
• 3(1)2 + 3(2)2 + ….
3(500)2
99
n
n 1
25
 2n
n 0
500
2
3
n

n 1
Definite Integrals
Lesson 5.2
Objectives
• Students will be able to
– express the area under a curve as a definite integral
and as a limit of Riemann sums.
– compute the area under a curve using a numerical
integration procedure.
Key Concept: Riemann Sum
A Riemann sum, Rn, for function f on the interval [a, b]
is a sum of the form
n
Rn   f ck   xk
k 1
where the interval [a, b] is partitioned into n
subintervals of widths Δxk, and the numbers {ck} are
sample points, one in each subinterval.
Example: Calculating Riemann Sums
a. Upper = using right endpoints: ¼ ( 1/8 + 27/64 + 1) = 99/256
b. Lower: using left endpoints: ¼ ( 1/64 + 1/8 + 27/64) = 9/64
Definite Integral
Upper limit
of integration
n
lim
n 
Integral
sign
The function
is the
integrand
b
 f (c )x   f ( x)d x
k 1
k
a
x is the
variable of
integration
lower limit of
integration
Let f be continuous on [a,b] and be partitioned into n
subintervals of equal length Δx = (b – a)/n
ck is some point in the kth subinterval.
When you find the value of the integral, you have evaluated the
integral
Express each limit as a definite
interval
2
n
lim
n 
c
k 1
2
k
x
x, 0,2
n
n 
 (c
k 1
dx.
0
5
lim
2
2
k
 3ck )x,  7,5
2
(
x
  3x) dx.
7
The Integral of a Constant
b
b
a
a
 f ( x)dx  cdx c(b  a)
1
 5dx 5(1  (2))  5(3)  15
2
Key Concept: Area Under a Curve
If y = f (x) is nonnegative and integrable on [a, b], and
if Rn is any Riemann sum for f on [a, b], then
b
lim Rn   f x  dx.
x 0
a
Use the graph of the integrand and
areas to evaluate the integral
4
 ( x / 2  3) dx.
2
Homework
• Day 1: p. 282/3: 3-6, 8-12, 14, 29-30
• Day 2: p. 283: 16-22, 33-36 (NINT means fnInt
on your calculator)
Drill (let h be measured in feet)
h(t) = -5t2 + 20t + 15
• a. Estimate the
instantaneous velocity
of the ball 3 seconds
after it’s thrown.
• b. Estimate the
acceleration of the ball
3 seconds after it’s
thrown.
• c. Estimate the
maximum height.
• v(t) = -10t + 20
• v(3) = -30 + 20 = -10 ft/s
• a(t) = -10 ft/ s2
• 0 = -10t + 20
• t=2
• h(2) = 35 feet
Example Area Under a Curve
Determine the area under the curve over the interval
[–4, 4].
y  16  x
1 2
A  r
2
1
2
A   4 
2
A  8
2
Example Area Under a Curve
Evaluate the integral.

4
4
16  x dx  8
2
3
Example: Estimate 
9  x 2 dx
2
• Solution: Graph the function. It is a semi-circle. The shaded
area represents the area of the region bounded by the
semicircle, the x-axis, and the lines x =2 and x = 3.
• Determine the area of the sector of the circle: ½r2θ, where r is
radius and θ is the angle in RADIANS!
• You will need to find θ by tan-1(y/x)
• Determine the area of the triangle: ½bh or ½ xy
• Subtract the two areas.
Θ
•
•
•
•
r=3
Θ= tan-1( 5 /2)=
.84 radians
Θ
½r2θ = 3.78
• A = ½ (2) 5 = 2.24
• 3.78-2.24 = 1.54
(x,y) = (2, 9  2 2 ) = (2, 5 )
5
Θ
3

2
9  x 2 dx
Using your calculator to determine
integrals
• Math
• 9: fnInt
• fnInt(f(x), x , a, b)
Example Using Your Calculator to
Evaluate Integrals
Approximate the following integrals using a calculator.

12

144  x  1 dx  125.09738
2
0
5

dx

 12
5.566371
1  4

1
4
 x  3 dx  2.55003812
1