The Definite Integral as Area
November 6, 2013
The Definite Integral as Area
The case when f (x) is nonnegative
Let f (x) is nonnegative and continuous for a ≤ x ≤ b.
The Definite Integral as Area
The case when f (x) is nonnegative
Let f (x) is nonnegative and continuous for a ≤ x ≤ b.
The left- and right-hand sums tends to the area under the
graph of f between a and b as ∆t smaller and smaller.
The Definite Integral as Area
The case when f (x) is nonnegative
Let f (x) is nonnegative and continuous for a ≤ x ≤ b.
The left- and right-hand sums tends to the area under the
graph of f between aRand b as ∆t smaller and smaller.
b
The definite integral a f (x)dx is the area.
The Definite Integral as Area
The case when f (x) is nonnegative
Let f (x) is nonnegative and continuous for a ≤ x ≤ b.
The left- and right-hand sums tends to the area under the
graph of f between aRand b as ∆t smaller and smaller.
b
The definite integral a f (x)dx is the area.
The Definite Integral as Area
The case when f (x) is nonnegative
Let f (x) is nonnegative and continuous for a ≤ x ≤ b.
The left- and right-hand sums tends to the area under the
graph of f between aRand b as ∆t smaller and smaller.
b
The definite integral a f (x)dx is the area.
The Definite Integral as Area
Example 1
Find the area under the graph of y = 10x(3−x ) between x = 0 and
x = 3.
The Definite Integral as Area
Example 1
Find the area under the graph of y = 10x(3−x ) between x = 0 and
x = 3.
The Definite Integral as Area
Example 1
Find the area under the graph of y = 10x(3−x ) between x = 0 and
x = 3.
Since theRfunction is nonnegative for 0 ≤ x ≤ 3, the area is
3
equal to 0 10x(3−x )dx.
The Definite Integral as Area
Example 1
Find the area under the graph of y = 10x(3−x ) between x = 0 and
x = 3.
Since theRfunction is nonnegative for 0 ≤ x ≤ 3, the area is
3
equal to 0 10x(3−x )dx.
Using calculator or computer, we can obtain
Z 3
10x(3−x )dx = 6.967
0
The Definite Integral as Area
Example 1
Find the area under the graph of y = 10x(3−x ) between x = 0 and
x =3
Since theRfunction is nonnegative for 0 ≤ x ≤ 3, the area is
3
equal to 0 10x(3−x )dx.
Using calculator or computer, we can obtain
Z 3
10x(3−x )dx = 6.967
0
See the calculator manual on Oncourse.
The Definite Integral as Area
Example 1
Find the area under the graph of y = 10x(3−x ) between x = 0 and
x =3
Since theRfunction is nonnegative for 0 ≤ x ≤ 3, the area is
3
equal to 0 10x(3−x )dx.
Using calculator or computer, we can obtain
Z 3
10x(3−x )dx = 6.967
0
See the calculator manual on Oncourse.
Use math softwares, such as Maple, Mathlab,
The Definite Integral as Area
Example 1
Find the area under the graph of y = 10x(3−x ) between x = 0 and
x =3
Since theRfunction is nonnegative for 0 ≤ x ≤ 3, the area is
3
equal to 0 10x(3−x )dx.
Using calculator or computer, we can obtain
Z 3
10x(3−x )dx = 6.967
0
See the calculator manual on Oncourse.
Use math softwares, such as Maple, Mathlab,
Online free resource:
http://www.numberempire.com/definiteintegralcalculator.php
The Definite Integral as Area
Relationship Between Definite Integral and Area : When
f (x) is NOT nonnegative.
Consider
R 1 2 the relationship between the definite integral 2
−1 (x − 1)dx and the area between the parabola y = x − 1 and
the x-axis.
The Definite Integral as Area
Relationship Between Definite Integral and Area : When
f (x) is NOT nonnegative.
Consider
R 1 2 the relationship between the definite integral 2
−1 (x − 1)dx and the area between the parabola y = x − 1 and
the x-axis.
The Definite Integral as Area
Relationship Between Definite Integral and Area : When
f (x) is NOT nonnegative.
The parabola lies below the x-axis between x = −1 and x = 1.
The Definite Integral as Area
Relationship Between Definite Integral and Area : When
f (x) is NOT nonnegative.
The
lies below the x-axis between x = −1 and x = 1.
R 1 parabola
2
−1 (x − 1)dx = −Area = −1.33
The Definite Integral as Area
Conclusion
When f (x) is positive for some x-values and negative for others,
and
R b a < b:
a f (x)dx is the sum of the areas above the x-axis, counted
positively, and the areas below the x-axis, counted negatively.
The Definite Integral as Area
Example 3
Interpret the definite integral
areas.
R4
0
(x 3 − 7x 2 + 11x)dx in terms of
The Definite Integral as Area
Example 3
Interpret the definite integral
areas.
R4
0
(x 3 − 7x 2 + 11x)dx in terms of
The Definite Integral as Area
Example 3
the graph of f (x) is crossing below the x-axis about x = 2.4.
The Definite Integral as Area
Example 3
the graph of f (x) is crossing below the x-axis about x = 2.4.
R4
Use calculator 0 (x 3 − 7x 2 + 11x)dx = 2.67
The Definite Integral as Area
Example 3
the graph of f (x) is crossing below the x-axis about x = 2.4.
R4
Use calculator 0 (x 3 − 7x 2 + 11x)dx = 2.67
R4 3
2
0 (x − 7x + 11x)dx = A1 − A2 .
The Definite Integral as Area
Example 3
the graph of f (x) is crossing below the x-axis about x = 2.4.
The Definite Integral as Area
Example 3
the graph of f (x) is crossing below the x-axis about x = 2.4.
R4
Use calculator 0 (x 3 − 7x 2 + 11x)dx = 2.67
The Definite Integral as Area
Example 3
the graph of f (x) is crossing below the x-axis about x = 2.4.
R4
Use calculator 0 (x 3 − 7x 2 + 11x)dx = 2.67
R4 3
2
0 (x − 7x + 11x)dx = A1 − A2 .
The Definite Integral as Area
Example 3
the graph of f (x) is crossing below the x-axis about x = 2.4.
R4
Use calculator 0 (x 3 − 7x 2 + 11x)dx = 2.67
R4 3
2
0 (x − 7x + 11x)dx = A1 − A2 .
BreakRthe integral into two parts
2.4
A1 ≈ 0 (x 3 − 7x 2 + 11x)dx ≈ 7.72 and
R4
−A2 ≈ 2.4 (x 3 − 7x 2 + 11x)dx ≈ −5.05.
The Definite Integral as Area
Property
Assume f (x) is continuous from a to b, and a < c < b. Then
Z
b
Z
f (x)dx =
a
c
Z
f (x)dx +
a
b
f (x)dx
c
The Definite Integral as Area
Example 4
For each functions graphed in the figure, decide whether
is positive, negative or approximately zero.
The Definite Integral as Area
R5
0
f (x)dx
Example 4
For each functions graphed in the figure, decide whether
is positive, negative or approximately zero.
The Definite Integral as Area
R5
0
f (x)dx
Example 4
For each functions graphed in the figure, decide whether
is positive, negative or approximately zero.
The Definite Integral as Area
R5
0
f (x)dx
Area between two curves
The Definite Integral as Area
Area between two curves
If g (x) ≤ f (x) for a ≤ x ≤ b. Then
Area between graphs of f (x) and g (x) for a ≤ x ≤ b is equal to
Rb
a (f (x) − g (x))dx.
The Definite Integral as Area
Formula
Assume f (x) and g (x) are two continuous functions for a ≤ x ≤ b,
and A and B are two constants. Then
Z b
Z b
Z b
(Af (x) + Bg (x))dx = A
f (x)dx + B
g (x)dx
a
a
a
The Definite Integral as Area
Example 5
Graph f (x) = 4x − x 2 and g (x) = 21 x 3/2 for x ≥ 0. Use a definite
integral to estimate the area enclosed by the graphs of these two
functions.
The Definite Integral as Area
Example 5
Graph f (x) = 4x − x 2 and g (x) = 21 x 3/2 for x ≥ 0. Use a definite
integral to estimate the area enclosed by the graphs of these two
functions.
The Definite Integral as Area
Example 5
Graph f (x) = 4x − x 2 and g (x) = 21 x 3/2 for x ≥ 0. Use a definite
integral to estimate the area enclosed by the graphs of these two
functions.
Two graph cross at x = 0 and x ≈ 3.1.
The Definite Integral as Area
Example 5
Graph f (x) = 4x − x 2 and g (x) = 21 x 3/2 for x ≥ 0. Use a definite
integral to estimate the area enclosed by the graphs of these two
functions.
Two graph cross at x = 0 and x ≈ 3.1.
Between x = 0 and x = 3.1 the graph of f (x) = 4x − x 2 is
above the graph of g (x) = 12 x 3/2 .
The Definite Integral as Area
Example 5
Graph f (x) = 4x − x 2 and g (x) = 21 x 3/2 for x ≥ 0. Use a definite
integral to estimate the area enclosed by the graphs of these two
functions.
Two graph cross at x = 0 and x ≈ 3.1.
Between x = 0 and x = 3.1 the graph of f (x) = 4x − x 2 is
above the graph of g (x) = 12 x 3/2 .
R 3.1
The Area is (approximately) 0 (4x − x 2 − 12 x 3/2 )dx.
The Definite Integral as Area
Area between two curves
a≤x ≤b
RFor
b
(f
(x) − g (x))dx is the sum of the areas when f (x) > g (x),
a
counted positively, and the areas when f (x) < g (x), counted
negatively.
The Definite Integral as Area
Area between two curves
Z
10
(sin(x) − sin(−x))dx
0
The Definite Integral as Area