Calculus 2 formula Sheet
University: Назарбаев Университеті
Course: Calculus II (MATH 162)
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Students shared 11 documents in this course
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