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Arc Length and Surfaces of Revolution
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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Answer Example Find the surface area formed by revolving
around the x-axis from 1 to 4. Answer ~30.85
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Answer Example Find the surface area formed by revolving
around the y-axis from 0 to 2. Answer ~36.18
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