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Application of Integration (Area)
Bander Almutairi
King Saud University
November 17, 2015
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 1 / 14
1 Area
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 2 / 14
Area
If f (x) ≥ 0 and continuous on the interval [a, b],
then the definite integral∫ ba f (x)dx gives the area of the the region under the graph of f from a tob.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 3 / 14
Area
If f (x) ≥ 0 and continuous on the interval [a, b], then the definite integral∫ ba f (x)dx gives the area of the the region under the graph of f from a tob.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 3 / 14
Area
If f (x) ≥ 0 and continuous on the interval [a, b], then the definite integral∫ ba f (x)dx gives the area of the the region under the graph of f from a tob.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 3 / 14
Area
Example: Find the area under the graph y = x2 − 4x + 5 from 0 to 5.
Solution:The function y represents a parabola with a vertex at the point(2, 1),
A =
∫ 5
0(x2 − 4x + 5)dx =
50
3.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 4 / 14
Area
Example: Find the area under the graph y = x2 − 4x + 5 from 0 to 5.Solution:
The function y represents a parabola with a vertex at the point(2, 1),
A =
∫ 5
0(x2 − 4x + 5)dx =
50
3.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 4 / 14
Area
Example: Find the area under the graph y = x2 − 4x + 5 from 0 to 5.Solution:The function y represents a parabola with a vertex at the point(2, 1),
A
=
∫ 5
0(x2 − 4x + 5)dx =
50
3.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 4 / 14
Area
Example: Find the area under the graph y = x2 − 4x + 5 from 0 to 5.Solution:The function y represents a parabola with a vertex at the point(2, 1),
A =
∫ 5
0(x2 − 4x + 5)dx =
50
3.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 4 / 14
Area
Note that: If f (x) ≤ 0 for x ∈ [a, b],
then −f (x) ≥ 0, for x ∈ [a, b], sothe area above the graph is
A = −∫ b
af (x)dx
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 5 / 14
Area
Note that: If f (x) ≤ 0 for x ∈ [a, b], then −f (x) ≥ 0, for x ∈ [a, b],
sothe area above the graph is
A = −∫ b
af (x)dx
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 5 / 14
Area
Note that: If f (x) ≤ 0 for x ∈ [a, b], then −f (x) ≥ 0, for x ∈ [a, b], sothe area above the graph is
A = −∫ b
af (x)dx
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 5 / 14
Area
Note that: If f (x) ≤ 0 for x ∈ [a, b], then −f (x) ≥ 0, for x ∈ [a, b], sothe area above the graph is
A = −∫ b
af (x)dx
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 5 / 14
Area
Note that: If f (x) ≤ 0 for x ∈ [a, b], then −f (x) ≥ 0, for x ∈ [a, b], sothe area above the graph is
A = −∫ b
af (x)dx
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 5 / 14
Area
Theorem
If f and g are continuous and f (x) ≥ g(x) for x ∈ [a, b],
then the area Aof the region bounded by the graphs of f , g from x = a to x = b is
A =
∫ b
a[f (x)− g(x)] dx .
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 6 / 14
Area
Theorem
If f and g are continuous and f (x) ≥ g(x) for x ∈ [a, b], then the area Aof the region bounded by the graphs of f , g from x = a to x = b is
A =
∫ b
a[f (x)− g(x)] dx .
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 6 / 14
Area
Theorem
If f and g are continuous and f (x) ≥ g(x) for x ∈ [a, b], then the area Aof the region bounded by the graphs of f , g from x = a to x = b is
A =
∫ b
a[f (x)− g(x)] dx .
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 6 / 14
Area
Exercise: Express the area between the graphs in the following diagram asa definite integral:
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 7 / 14
Area
Example: (1) Sketch the region bounded by graphs y =√x and y = x2,
and find its area.
Solution:The area is
A =
∫ 1
0
[√x − x2
]dx =
1
3.
(2) Find the area of the region bounded by the graphes y − x = 6, y = x3
and 2y + x = 0.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 8 / 14
Area
Example: (1) Sketch the region bounded by graphs y =√x and y = x2,
and find its area.Solution:
The area is
A =
∫ 1
0
[√x − x2
]dx =
1
3.
(2) Find the area of the region bounded by the graphes y − x = 6, y = x3
and 2y + x = 0.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 8 / 14
Area
Example: (1) Sketch the region bounded by graphs y =√x and y = x2,
and find its area.Solution:The area is
A =
∫ 1
0
[√x − x2
]dx =
1
3.
(2) Find the area of the region bounded by the graphes y − x = 6, y = x3
and 2y + x = 0.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 8 / 14
Area
Example: (1) Sketch the region bounded by graphs y =√x and y = x2,
and find its area.Solution:The area is
A =
∫ 1
0
[√x − x2
]dx =
1
3.
(2) Find the area of the region bounded by the graphes y − x = 6, y = x3
and 2y + x = 0.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 8 / 14
Area
Example: (1) Sketch the region bounded by graphs y =√x and y = x2,
and find its area.Solution:The area is
A =
∫ 1
0
[√x − x2
]dx =
1
3.
(2) Find the area of the region bounded by the graphes y − x = 6, y = x3
and 2y + x = 0.
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 8 / 14
Area
Exercise: Find the shaded area for the following diagrams:[1]
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 9 / 14
Area
[2]
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 10 / 14
Area
[3]
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 11 / 14
Area
[4]
Bander Almutairi (King Saud University) Application of Integration (Area) November 17, 2015 12 / 14