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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