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Math 152, Benjamin
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7.1 Area Between Curves
If y = f (x) and y = g(x) are continuous functions and if f (x) ≥ g(x) on an interval a ≤ x ≤ b, then the
area of the region bounded by the curves on that interval is
Z
b
[f (x) − g(x)] dx
a
If x = f (y) and x = g(y) are continuous functions and if f (y) ≥ g(y) on an interval a ≤ y ≤ b, then the
area of the region bounded by the curves on that interval is
Z
b
a
[f (y) − g(y)] dy
Examples: Find the area of the region(s) bounded by the curves.
• y = 5x − x2 , y = 2x − 4
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Math 152, Benjamin
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• x = y 2 , x − 2y = 3
• y = |x| − 1, y = 5 − x2
• x = 10 − y 2 , x = 2y 2 − 5
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• y = x2 + 4, y = 2x2 , from x = −2 to x = 4.
• y = 1 − sin x, y = sin x from x = 0 to x =
π
2
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Math 152, Benjamin
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Set up the integral(s) to find the area bounded by the curves y = 1/x2 , y = x, and y = 18 x.
Find the area of the region bounded by the curve y = 2x2 , the tangent line to this curve at the point (3, 18),
and the x-axis.
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7.2 Volumes
General Principle:
Disks
About x-axis:
About y-axis:
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Examples: Find the volumes of the solids found by rotating the regions bounded by the given curves about
the specified axis/line.
√
• y = 2 x, y = 0, x = 9 about the x-axis
• y = x3 , y = 8, x = 0 about the y-axis
• y = (x − 2)2 , y = 9 about the line y = 9
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Math 152, Benjamin
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• x = (y − 1)2 + 2, x = 6 about the line x = 6
Washers
Examples: Find the volumes of the solids found by rotating the regions bounded the given curves about the
specified axis/line.
• y = x2 + 2, y = 3x about the x-axis
• Same region about the y-axis
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• x = y 2 , y = x3 about the line y = 3
• Same region about the line x = −2
•y=
√
x + 1, y = 1, x = 4 about the line x = 8
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Same region about the line y = −5:
Volumes by Cross-Sections
Example where base is a circle and cross-sections perpendicular to the x-axis are equilateral triangles.
We’ll do a problem like this in the third example below. First, we’ll do a few ”easier” ones.
Example 1: Find the volume of the solid S whose base is the region bounded by the curves y = x2 and
y = 4. Cross sections perpendicular to the y-axis are squares.
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Example 2: Find the volume of the solid S whose base is the triangular region with endpoints (0, 0), (6, 0),
and (0, 2). Cross-sections perpendicular to the x-axis are semicircles.
Example 3: Find the volume of the solid S whose base is the circle x2 + y 2 = 9. Cross sections perpendicular
to the x-axis are equilateral triangles.
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