Arc Length and Surfaces of Revolution

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Lesson 7.4

• Arc Length and Surfaces of Revolution

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We want to be able to find the distance travelled
by an object along a curve This is the same as
finding the length of a curve from a given point
to another The integral we will use comes from
the infinite limit of the lengths of segments
along the curve
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Arc Length Along a Curve Distance, s, is given
by or If the function is
differentiable along the interval a, b
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Example Find the arc length on the interval 1, 2
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Area of a Surface of Revolution We are trying to
find the lateral area of the cylinder
below The formula for this is found using
circumference and length of the cylinder
circumference
Cylinder length
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The length, L, in this formula is the length of
the curve used earlier Then, applying an
infinite limit of infinitely small cylinders
along with a changing radius, we get the integral
for surface area of revolution If the length,
L, of the cylinder is vertical then
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Example Find the surface area formed by
revolving around the x-axis from 1 to 4.
30.85
Answer
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Example Find the surface area formed by
revolving around the y-axis from 0 to 2.
36.18
Answer