Integral Calculus Flashcards
True or false, indefinite integral has limits
False
∫ a du =
a ∫ du = au + c
∫ a^u du =
a^ u / In a + c , a > 1, a=1
∫ u^n du
= 1/ n +1 u^n+1
∫e^u du
= e^u + C
∫ u^-1 du
= du/u = ln abs u + c
∫ln u du
=u ln abs u - u +c
∫ sin u
= -cos u+ c
∫ cos u du
= sin u + c
∫ tan u du
= ln abs sec u + c
∫ cot u du
= ln abs sin u + c
∫ sec u du
= ln abs sec u + tan u + c
∫csc u du
= ln |csc u - cot u| + c
= - ln |csc u - cot u| + c
∫ sec u tan u du
= sec u + c
∫csc u cot u du
= - csc u + c
∫sec^2 u du
= tan u+ C
∫csc^2 u du
= -cot u + c
Indefinite Integral caltech
- Input given, let x be any uncommon number
- Differentiate all choices usng d/dx function in calcu, let x be same in step 1
- Which ever is the same in step 1 is the answer
Integral by parts caltech
Differentiate u and v till zero, see notes for full step
Plane areas: Step in solving rectangular strip
- Plot the given curve
- Choose what strip to use
- Solve for the equation to be integrated (Xr-Xl) or (Yu-Yl)
- Find the limits (POI), equate the two equations solved from step 3
Plane areas: Step in solving radial strip
- Make equation equal to r
- Substitute r to radial strip formula
- Get limits using mode 6 (table)
Start: 0, End: 2pi, Step: pi/12
Area of some polar curve: r2 = k cos 2 theta
Area= K
link to formula
Area of some polar curve: r2 = k sin 2 theta
Area= K
link to formula
Area of some polar curve: r2 = k sin theta
Area= 2K
link to formula
Area of some polar curve: r2 = k cos theta
Area= 2K
link to formula
Area of some polar curve: r = k(1+cos theta)
Area = 1.5 pi K^2
Perimeter = 2pi a
link to formula
Area of some polar curve: r = k(1+sin theta)
Area = 1.5 pi K^2
Perimeter = 2 pi a
link to formula
Area of some polar curve: r = k sin 3 theta
Area = 1/4 pi k^2
link to formula
Area of some polar curve: r = k cos 3 theta
Area = 1/4 pi k^2
link to formula
Area of some polar curve: r = k cos 2 theta
Area = 1/2 pi k^2
link to formula
Area of some polar curve: r = 2 k cos theta
Area = pi k^2
link to formula
Area of some polar curve: r = 2 k sin theta
Area = pi k^2
link to formula
Volume of Solid of Revolution: Disk method formula
In disk method: the orientation of the strip must be ______ to the axis of revolution
perpendicular
Volume of Solid of Revolution: Disk method formula
In ring method: the orientation of the strip must be ______ to the axis of revolution
perpendicular
Volume of Solid of Revolution: Shell method formula
Second Proposition of Pappus, what method and formula
Shell Method
V = 2pi A d
Length of Curves: Parametric Formula
Length of Curves: Rectangular Formula
Length of Curves: Polar Formula
First Proposition of Pappus, what method and formula
Surface area of Curves
SA = 2 pi integral of d.ds
First Moment of Area
Centroid
Centroid Formula
Second Moment of Area
Moment of Inertia
Moment of Inertia resists:
bending
Moment of Inertia: y- axis
Iy = integral of x^2 dA
or
Iy = 1/3 integral of x^3 dy
Moment of Inertia: x- axis
Ix = integral of y^2 dA
or
Ix = 1/3 integral of y^3 dx
Moment of Inertia: differential area must be _______ to the “with respect of x/y axis”
Parallel
Polar Moment of Inertia Formula
J= R^2 dA
or
J= Iy + Ix
Product of Inertia Formula
Ixy = integral of xy dA
Work Problems
Work = integral of f(x) . dX
