12. Applications

Calculus 2



Weiqi Zhou

Example

  • find the area of the shape surrounded by and

  • intersections are ，

  • is above for

  • area

Example

  • find the area of the shape surrounded by and ,

  • area of a sector:

  • area


  • remark: the shape is ，i.e., .

Exercises

  • find the area of the shape surronded by and

  • find the area of the ellipse

  • find the area of the shape surronded by , the axis, and a tagent line of that passes the origin

  • find the area of the shape surrounded by

Summation

  • finite sum:

  • countable sum：

  • integration is a way of weighted uncountable sum
    e.g., mass of an object with uniform density

Strategy: Split and Sum up

Example

  • find the volume of the solid obtained by rotating around the axis

  • the cross section is a disc of radius , its area is

  • volume

Exercise

a cylinder of radius is sected by a plane that passes the center of its base, the angle between the cross section and the base is , find the volume of the solid below the cross section

Example

  • find the perimeter of

  • ，

  • curve length

Example

  • find the perimeter of

  • the upper half: ，

  • curve length

Exercise

find the length of

Summary

  • Integration as Summation

  • Calculation of Areas

  • Calculation of Volumes

  • Calculation of Lengths