Calculus 2 formula Sheet

University: Назарбаев Университеті

Course: Calculus II (MATH 162)

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CalculusII. Formula sheet

by ME

January 20, 2021

6 Application of Definitie Integrals

6.2 Volume By slicing. Washers and Cylindrical shells

6.2.1 VOLUMES BY SLICING

Volume of figure bounded by two parallell planes perpendicular to x-axis.

V=Zb

a

A(x)dx

Volume of figure bounded by two parallel planes perpendicular to y-axis.

V=Zb

a

A(y)dy

Here A(x) is the equation of sliced area

6.2.2 VOLUMES BY DISKS PERPENDICULAR TO THE x-AXIS

Volume of solid made by revolution

V=Zb

a

π[f(x)]2dx

Volume of solid made by revolution with the hole in it

V=Zb

a

π([f(x)]2

−[g(x)]2)dx

6.2.3 VOLUMES BY DISKS AND WASHERS PERPENDICULAR TO THE y-AXIS

Disks

V=Zb

a

π[u(y)]2dy

Washers

V=Zb

a

π([w(y)]2

−[v(y)]2)dy

6.3 Volume by cylindrical shells

Volume by c.s. about y-axis

V=Zb

a

2πxf(x)dx

Or when figure is limited by two equations we simply substract one from another

V=Zb

a

2π[f(x)−g(x)] dx

Volume by c.s. about x-axis would follow same rules as disks and washers

V=Zb

a

2π[w(y)−u(y)] dy

6.5 Surface Area

If fis a smooth, nonegative function on [a, b], then the surface area Sof the surface of revolution that

is generated by revolving about x-axis is

S=Zb

a

2πf(x)q1 + [f′(x)]2dx

OR

1