Cross Sections, Shells, Disks, Washers etc

Steps for Area between curves

1. Draw picture
2. Find points of intersection
3. Set up integral

(Using Calculator)


4. Simplify
5. Integrate
6. Simplify

How do you set up your integral to find the area between curves?

Set up your integral from the points of intersection, then Take the function for the top curve minus the function for the bottom

ex ∫0to3 f(x)-g(x)

where f(x) is the top function and g(x) is the bottom, and 0 &3 are the x coordinates of the closed interval

Arc Length Formula

∫a to b √1 + \[f’(x)\]² dx

What is a cross section?

The intersection of a plane & a 3D figure

V = ∫a to b A(x)dx

when cross section is perpendicular to x-axis

Area of a square

s²

Area of a triangle

1/2bh

Area of an equilateral triangle

__√3__ (s²)

4

Area of a circle

πr²

Disk Formula

V= π ∫a to b \[R(x)\]² dx

What is R(x) in terms of disks and washers

\[ (the top function)-(the bottom function)\]

Washer Formula

π ∫ a to b \[ (R(x))² - (r(x))² \] dx

What is r(x) in terms of washers

The area of the hole

What is R(x) in terms of washers

The total area

Shell Formula

2π ∫ a to b p(x) h(x) dx

What is p(x) in terms of shells?

The distance from the center of your rectangle, to the axis of revolution

What is h(x) in terms of shells?

The height of your rectangle

For disks and washers, your rectangle should be _____ to the axis of revolution

perpendicular

For shells, your rectangle should be _____ to the axis of revolution

parallel

How do you draw the rectangle for disks and washers?

draw from the axis of revolution, into the boundaries of your functions until you’ve reached the end of your boundary

How do you draw the rectangle for shells?

Draw within the boundaries of your functions a rectangle parallel to the axis of revolution