1. 6.4-Length of a plane curve *Arc Length
If you were riding this function what would the distance of the ride be from x=1 to x=6?
Y=f(x)

2. Formula for Arc Length

3. Example:
What is the arc length of the given curve from (1,1) to (4,32)?

4. Solve this definite integral!

5. 6.5-Area of a surface of revolution *Surface Area
Y=f(x)
If we wanted to build objects of this shape how much material would we need to cover the surface?

6. Formula for Surface Area

7. Example of Surface Area
Find the area of the surface generated by revolving the given curve about the x-axis, from x=1 to x=4.

8. 6.7-Work The work done by a machine or person is given by the formula:
W=work, F=Force, D=Distance the force is applied.

9. Formula for Work
If F(x)=Variable force and [a,b] represents the interval the force is applied over, then the total work done is given by:

10. Example of an application of Work
Hooke’s Law gives us the force exerted by a spring at x units of displacement from its natural length.
K-The spring constant represents the stiffness of that spring. This is dependent upon the material the spring is made of and the tightness of the coils.

11. Example of a Work problem
A spring exerts a force of 6 N when stretched 1m beyond its natural length. How much work is required to stretch the spring 2m beyond its natural length?
We know F(x)=kx, so we must find K
We know F(1)=K(1)=k and F(1)=6, so k=6
We also know the string is being stretched from x=0 to x=2m.

12. Solution The work done is given by:
J stands for jules, which are units of work.

13. 6.8-Fluid Pressure and its’ Force
The relationship between pressure and force is given by:
P-Pressure, F-Force, A-Area of application on the surface

14. Force of a fluid
We need two concepts to generate a formula for the force applied by a fluid
Mass Density of the fluid δ: The mass per unit of volume.
Weight Density of the fluid ρ: The weight per unit of volume.

15. Force of a Fluid We can arrive at the conclusion that the force:
And the pressure:
Where h represents the vertical depth of the submersed object from the surface of the fluid!

16. Formula for Fluid Force
A flat surface immersed vertically in a fluid of density ρ, if the surface extends from x=a to x=b, w(x) is the width of the surface and h(x) is the depth, then the fluid force F is given by:
Note: This assumes we orient our usual x-axis in a vertical way so you can also calculate the fluid force by letting the y-axis represent the vertical in the same way.

17. Example of Force on a Vertical Surface
Example :   A 3 x 2 square window on the new Disney Cruise liner is to be built so that top of the window is four feet below the surface of the water.  What total force will the window be subjected to?  
4ft y Δy

18. Hence the total force from the top of the window to the bottom is:
Solution:   We take horizontal cross sections.  Letting y be the distance from the top of the window to the cross section and the weight density of salt water to be 62.4, we have:
 ΔF  =  62.4(4 + y)(3)(Δy)
Hence the total force from the top of the window to the bottom is: