The document discusses how to calculate the area under a curve using definite integrals and the Fundamental Theorem of Calculus. It explains that the area can be approximated as the sum of rectangles and becomes exact as the width approaches zero. The area is then given by the definite integral from a to b of the function, which is equal to evaluating the antiderivative at b and subtracting the evaluation at a. Examples demonstrate calculating areas under parabolic and exponential curves using this process.

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

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

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

4. Area Under the Curve
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.

5. Area Under the Curve
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.

6. Area Under the Curve
Find the area of the shaded region.

7. Area Under the Curve
= 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.

8. Area Under the Curve
=  
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.

9. Area Under the Curve
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.

10. 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.

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

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

13. The Fundamental Theorem of Calculus
 
Consider f x x


14. The Fundamental Theorem of Calculus
 
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


15. The Fundamental Theorem of Calculus
 
Consider f x x

Find the area between
the graph of f and the
x-axis on the interval
[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


16. The Fundamental Theorem of Calculus
  2
Consider f x x


17. The Fundamental Theorem of Calculus
  2
Consider f x x

Find the area between
the graph of f and the
x-axis on the interval
[0, 1].
1
2
0
Area x dx
 
3
3
x

0
1
 
1
3
  
0
3

2
1
3
Area units


18. The Fundamental Theorem of Calculus
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 

19. The Fundamental Theorem of Calculus
 
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

20. 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.