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Ch 8.2 Day 2 Wkst AP Calc BC Name:

Arc Length and Surface Area

Objective: to use integrals to calculate arc length and surface area.

For a function, [pic], Arc Length =[pic].

For a function, [pic], rotated about the x-axis, Surface Area =[pic].

For parametric equations, [pic] and [pic], Arc Length = Distance Traveled

= [pic].

= [pic].

Example 1: Find the arc length of [pic] from x = 0 to x = 2 [pic]

Example 2: Find the surface area of the solid obtained from rotating [pic] about the x-axis from x = 0 to x = 2 .

Example 3: Find the length of the parametric curve for [pic] and [pic] for [pic].

Ch 8.2 Day 2 Wkst AP Calc BC Name:

Arc Length and Surface Area

For a function, [pic], Arc Length =[pic].

For a function, [pic], rotated about the x-axis, Surface Area =[pic].

For parametric equations, [pic] and [pic], Arc Length = Distance Traveled

= [pic].

= [pic].

Find the arc length of the graph of the function from x = 0 to x = 2 .

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. Consider the ellipse [pic].

a) Find the circumference the ellipse. b) Find the area of the ellipse.

6.

7. Find the surface area of the solid obtained from rotating [pic] about the x-axis from ( 1 , 1 ) to ( 4 , 2 ).

8. Find the surface area of the solid obtained from rotating [pic] about the x-axis from x = 1 to x = 2 .

9. Find the surface area if the arc from ( -3 , 4 ) to ( 3 , 4 ) of the equation, [pic], is revolved about the x-axis.

For problems #10 to #12, find the length of the parametric curves.

10. [pic] and [pic] for [pic]. NO CALCULATOR.

11. [pic] and [pic] for [pic]. NO CALCULATOR.

12. [pic] and [pic] for [pic]. CALCULATOR REQUIRED.

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Not an AP Topic

Not an AP Topic

[pic]

ANSWERS:

1) 3.526 7) 30.846

2) À 8) 51.882 (or 16,911 À/1024)

3) sinh 2 (or 3.627) 9) 188.496

4) 0.733 10) [pic]

5a) 15.865 11) e - 1

b)π 8) 51.882 (or 16,911 π/1024)

3) sinh 2 (or 3.627) 9) 188.496

4) 0.733 10) [pic]

5a) 15.865 11) e - 1

b) 18.850 (or 6π) 12) 5.180

6)

[pic]

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