Areas and Definite Integrals.ppt

Areas & Definite Integrals
TS: Explicitly assessing
information and drawing
conclusions

Objectives
 To develop a formula for finding the area
under a curve.

Area Under the Curve
How do we find areas under a curve,
but above the x-axis?

14
1 2 13 14
1
Area ... i
i
R R R R R

      
As the number of rectangles used to approximate the area of
the region increases, the approximation becomes more accurate.

It is possible to find the exact area by letting the
width of each rectangle approach zero. Doing this
generates an infinite number of rectangles.

Find the area of the shaded region.

= um (height)  (base)
Area  sum of the areas of the rectangles
=  
f x x

=  
f x dx
a
b
The formula looks
like an integral.

=  
f x dx
a
b
The formula looks
like an integral.
Is the area really given by the antiderivative?
Yes!
The definite integral of f from a to b is the limit of the Riemann
sum as the lengths of the subintervals approach zero.

A function and the equation for the
area between its graph and the x-axis
are related to each other by the antiderivative.
=  
f x dx
a
b
The formula looks
like an integral.

Two Questions of Calculus
Q1: How do you find instantaneous velocity?
A: Use the derivative.
Q2: How do you find the area of exotic shapes?
A: Use the antiderivative.

How do we calculate areas under a curve,
but above the x-axis?

The Fundamental Theorem of Calculus
 
Area
B
A
f x dx
     
F B F A
 
   
where F x f x
 

 
Consider f x x


 
Consider f x x

Find the area between
the graph of f and the
x-axis on the interval
[0, 3].
1
2
Area bh

2
9
2
Area units


 
Consider f x x

[0, 3].
3
0
Area x dx
 
2
2
x
 C

0
3
The bar tells you to evaluate
the expression at 3 and
subtract the value of the
expression at 0.
 
9
2 C
   
0
2 C
 
2
9
2
Area units


  2
Consider f x x


  2
Consider f x x

[0, 1].
1
2
0
Area x dx
 
3
3
x

0
1
 
1
3
  
0
3

2
1
3
Area units


1
2
0
Evaluate 1
x x dx
 

2
Let 1
u x
 
 
1
2
2
1
x x dx
 

2
du x dx

 
1
2
x u dx 

1
2
u x dx
 

1
2 1
2
u du 

3
2
3
2
u
C
 
3
2
1
3 u C
 
Substitute into
the integral.
Always express your
answer in terms of the
original variable.
1
2 du x dx

1
2
1
2 u du 

1
2 

 
3/2
1
3 2
  
3/2
1
3 1

.609

 
3
2
2
1
3 1
x 
0
1
1
2
0
Evaluate 1
x x dx
 

So this would represent
the area between the curve
y = x √(x2 + 1) and the
x-axis from x = 0 to 1

Conclusion
 As the number of rectangles used to approximate the area of the
region increases, the approximation becomes more accurate.
 It is possible to find the exact area by letting the width of each
rectangle approach zero. Doing this generates an infinite number of
rectangles.
 A function and the equation for the area between its graph and the
x-axis are related to each other by the antiderivative.
 The Fundamental Theorem of Calculus enables us to evaluate
definite integrals. This empowers us to find the area between a
curve and the x-axis.

Areas and Definite Integrals.ppt

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Areas and Definite Integrals.ppt