1. 1.) Set up the integral to find the area of the shaded region
Chapter 6.4
Arc Length
1.) Set up the integral to find the area of the shaded region
2.) Draw a sketch of and set up the integral to find the volume of the
solid created when revolving the area around the x-axis using both
the washer and shell method.
3.) Draw a sketch of and set up the integral to find the volume of the
Solid created when revolving the area around the y-axis using both
4.) Find the perimeter of the region.
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2. Chapter 6.4
Arc Length
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3. Chapter 6.4
Arc Length
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4. Chapter 6.4
Arc Length
Find the arc length of the graph of the function over the indicated interval.
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5. Example:

6. If you have an equation that is easier to solve for x than for y, the length of the curve can be found the same way.
Notice that x and y are reversed.

7. Find the Perimeter of the shaded region in terms of k.
Chapter 6.4
Arc Length
Find the Perimeter of the shaded region in terms of k.
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8. Chapter 6.4
Arc Length
Practice:
P #1-5, 11-15, 17
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9. r Surface Area: Consider a curve rotated about the x-axis:
The surface area of this band is: r
The radius is the y-value of the function, so the whole area is given by:
This is the same ds that we had in the “length of curve” formula, so the formula becomes:
Surface Area about x-axis (Cartesian):
To rotate about the y-axis, just reverse x and y in the formula!

10. Example:
Rotate about the y-axis.

11. Example:
Rotate about the y-axis.

12. Example:
Rotate about the y-axis.
From geometry:

13. Example:
rotated about x-axis.

15. Homework:
447 #6, 21, 31, 33, 35, 37, 38
Also create a cheat sheet that organizes the formulas for 6.1 to 6.4