Areas, Volumes, Rotations

Question 1

What is the formula for the area under a curve ? Between 2 curves? What’s the top/side rules? What are the bounds ?

Answer 1

The integral of f(x)

The integral of (f(x)-g(x))

Top - bottom

Right - left

The bounds are found by setting the functions equal to each other and solving for the variable

Question 2

Trapezoidal Rule : and what is delta x equal to ?

Answer 2

Tn= delta (x) /2 * ( fx + 2fx1+2fx2+…fn)

Where delta x = b-a/ n

Question 3

Midpoint rule

Answer 3

Mn = delta x ( fx+fx+fx…fn)

Question 4

What is the formula for volumes by disks? Disk and washer forms!

Answer 4

Disk:
S (a->b) 🥧 r^2(variable)

Where r is really a function

Washer:

S (a->b) 🥧(R^2-r^2)

Where it’s the outer r - small r
Which is really outer function minus inner function

Question 5

What is the formula for volumes by shells?

Answer 5

S (a->b) 2 🥧 x* f(x) dx Or in terms of y

Question 6

What are the rules on r and H for volumes by shells and what does each symbol equate to in formula

Answer 6

R= the x or y variable
H= the f(x) or f(y) variable

R is always perpendicular to the axis of rotation

H is always parallel to the axis of rotation

Question 7

Formula for arc length

Answer 7

S (a->b) ROOT ( 1+ f ‘ (x) ^2 ) dx

Question 8

What is the area of a revolution

Answer 8

S (a->b) 2🥧f(x) * ROOT( 1+ f ‘ (x)^2 ) dx

Which is basically the integral of 2🥧f(x) times the arc length