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MAT137 | Calculus! | Lecture 30
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§6.1 More on Areas§6.2 Volumes
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§6.2 - 6.3 More on VolumesMethods of Integration
official website http://uoft.me/MAT137
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Area between Two Curves
Area between Two Curves
If f and g are integrable functions, the the the area between the curvesy = f (x) and y = g(x) from x = a to x = b is given by
∫b
a∣f (x) − g(x)∣dx
The absolute value appears in this formula because
∣f (x) − g(x)∣ =⎧⎪⎪⎨⎪⎪⎩
f (x) − g(x) if f (x) ≥ g(x),g(x) − f (x) if f (x) ≤ g(x).
π4
f
g
A
x
y
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Area between Two Curves
Example 1
Find the area of the region enclosed by the line x − y = 2 and the parabolax = y2.
0 1 2 3 4
−3
−2
−1
1
2
3
(1,−1)
(4,2)x = y 2
y = x − 2
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Area between Two Curves
Example 1
Find the area of the region enclosed by the line x − y = 2 and the parabolax = y2.
0 1 2 3 4
−3
−2
−1
1
2
3
(1,−1)
(4,2)x = y 2
y = x − 2
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Area between Two Curves
Example 1
Find the area of the region enclosed by the line x − y = 2 and the parabolax = y2.
0 1 2 3 4
−3
−2
−1
1
2
3
(1,−1)
(4,2)y =√x
y = −√x
y = x − 2
A1 A2
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Area between Two Curves
Example 2
Find the area of the region enclosed by the line x − y = 2 and the parabolax = y2.
0 1 2 3 4
−3
−2
−1
1
2
3
(1,−1)
(4,2)x = y 2
x = y + 2
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
Volume of Familiar Solids
What is the volume of the following solids?
r
h
V = πr2h
z
xy
V = xyx
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
Volume of Familiar Solids
What is the volume of the following solids?
r
h
V = πr2h
z
xy
V = xyx
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
Volume of Familiar Solids
What is the volume of the following solids?
r
h
V = πr2h
z
xy
V = xyx
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
Volume of Familiar Solids
What is the volume of the following solids?
r
h
V = πr2h
z
xy
V = xyx
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
What about the volume of the following solid?
V = area(B) ⋅ hBeatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumes
V ≈n
∑i=1
A(x∗i )∆x
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volumen
V = ∫b
aA(x)dx
where A(x) is the cross-sectional area as a function of x .
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volume
Example 2
Find the volume of the right circular cone with radius R and height h.
R
h
r
x
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volume
Example 3
The region R enclosed by the curves y = x and y = x2 is rotated about thex-axis. Find the volume of the resulting solid.
1
y = x2
y = x
x
y
(1,1)
R
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volume
Example 3
The region R enclosed by the curves y = x and y = x2 is rotated about thex-axis. Find the volume of the resulting solid.
1
y = x2
y = x
x
y
(1,1)
R
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
Volume
Example 4
The region R enclosed by the curves y = x and y = x2 is rotated about thex-axis. Find the volume of the resulting solid.
1
y = x2
y = x
x
y
(1,1)
R
A(x)
x2
x
Cross-section
Beatriz Navarro-Lameda L0601 MAT137 24 January 2017
