Integral Calculus

Question 1

True or false, indefinite integral has limits

Answer 1

False

Question 2

∫ a du =

Answer 2

a ∫ du = au + c

Question 3

∫ a^u du =

Answer 3

a^ u / In a + c , a > 1, a=1

Question 4

∫ u^n du

Answer 4

= 1/ n +1 u^n+1

Question 5

∫e^u du

Answer 5

= e^u + C

Question 6

∫ u^-1 du

Answer 6

= du/u = ln abs u + c

Question 7

∫ln u du

Answer 7

=u ln abs u - u +c

Question 8

∫ sin u

Answer 8

= -cos u+ c

Question 9

∫ cos u du

Answer 9

= sin u + c

Question 10

∫ tan u du

Answer 10

= ln abs sec u + c

Question 11

∫ cot u du

Answer 11

= ln abs sin u + c

Question 12

∫ sec u du

Answer 12

= ln abs sec u + tan u + c

Question 13

∫csc u du

Answer 13

= ln |csc u - cot u| + c
= - ln |csc u - cot u| + c

Question 14

∫ sec u tan u du

Answer 14

= sec u + c

Question 15

∫csc u cot u du

Answer 15

= - csc u + c

Question 16

∫sec^2 u du

Answer 16

= tan u+ C

Question 17

∫csc^2 u du

Answer 17

= -cot u + c

Question 18

Indefinite Integral caltech

Answer 18

1. Input given, let x be any uncommon number
2. Differentiate all choices usng d/dx function in calcu, let x be same in step 1
3. Which ever is the same in step 1 is the answer

Question 19

Integral by parts caltech

Answer 19

Differentiate u and v till zero, see notes for full step

Question 20

Plane Areas: Vertical Strip Formula

Answer 20

link to formula

Question 21

Plane Areas: Horizontal Strip Formula

Answer 21

link to formula

Question 22

Plane Areas: Radial Strip Formula

Answer 22

link to formula

Question 23

Plane areas: Step in solving rectangular strip

Answer 23

1. Plot the given curve
2. Choose what strip to use
3. Solve for the equation to be integrated (Xr-Xl) or (Yu-Yl)
4. Find the limits (POI), equate the two equations solved from step 3

Question 24

Plane areas: Step in solving radial strip

Answer 24

1. Make equation equal to r
2. Substitute r to radial strip formula
3. Get limits using mode 6 (table)
   Start: 0, End: 2pi, Step: pi/12

Question 25

Area of some polar curve: r2 = k cos 2 theta

Answer 25

Area= K [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 26

Area of some polar curve: r2 = k sin 2 theta

Answer 26

Area= K [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 27

Area of some polar curve: r2 = k sin theta

Answer 27

Area= 2K [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 28

Area of some polar curve: r2 = k cos theta

Answer 28

Area= 2K [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 29

Area of some polar curve: r = k(1+cos theta)

Answer 29

Area = 1.5 pi K^2 Perimeter = 2pi a [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 30

Area of some polar curve: r = k(1+sin theta)

Answer 30

Area = 1.5 pi K^2 Perimeter = 2 pi a [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 31

Area of some polar curve: r = k sin 3 theta

Answer 31

Area = 1/4 pi k^2 [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 32

Area of some polar curve: r = k cos 3 theta

Answer 32

Area = 1/4 pi k^2 [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 33

Area of some polar curve: r = k cos 2 theta

Answer 33

Area = 1/2 pi k^2 [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 34

Area of some polar curve: r = 2 k cos theta

Answer 34

Area = pi k^2 [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 35

Area of some polar curve: r = 2 k sin theta

Answer 35

Area = pi k^2 [link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 36

Volume of Solid of Revolution: Disk method formula

Answer 36

[link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 37

In disk method: the orientation of the strip must be ______ to the axis of revolution

Answer 37

perpendicular

Question 38

Volume of Solid of Revolution: Disk method formula

Answer 38

[link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 39

In ring method: the orientation of the strip must be ______ to the axis of revolution

Answer 39

perpendicular

Question 40

Volume of Solid of Revolution: Shell method formula

Answer 40

[link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 41

Second Proposition of Pappus, what method and formula

Answer 41

Shell Method V = 2pi A d

Question 42

Length of Curves: Parametric Formula

Answer 42

[link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 43

Length of Curves: Rectangular Formula

Answer 43

[link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 44

Length of Curves: Polar Formula

Answer 44

[link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 45

First Proposition of Pappus, what method and formula

Answer 45

Surface area of Curves SA = 2 pi integral of d.ds

Question 46

First Moment of Area

Answer 46

Centroid

Question 47

Centroid Formula

Answer 47

[link to formula](https://drive.google.com/file/d/15DTcnGslisKr_Q1jZvQCPRtYLLCPbY5M/view?usp=sharing)

Question 48

Second Moment of Area

Answer 48

Moment of Inertia

Question 49

Moment of Inertia resists:

Answer 49

bending

Question 50

Moment of Inertia: y- axis

Answer 50

Iy = integral of x^2 dA or Iy = 1/3 integral of x^3 dy

Question 51

Moment of Inertia: x- axis

Answer 51

Ix = integral of y^2 dA or Ix = 1/3 integral of y^3 dx

Question 52

Moment of Inertia: differential area must be _______ to the "with respect of x/y axis"

Answer 52

Parallel

Question 53

Polar Moment of Inertia Formula

Answer 53

J= R^2 dA or J= Iy + Ix

Question 54

Product of Inertia Formula

Answer 54

Ixy = integral of xy dA

Question 55

Work Problems

Answer 55

Work = integral of f(x) . dX