Applications of Integrals

Plane Area

Vertical Element

∫(Ya - Yb)dx

Ya = y-above Yb = y-below (in terms of x)

limits are x-coordinates from left to right

Plane Area

Horizontal Element

∫(Xr - Xl)dy

Xr = x to the right Xl = x to the left (in terms of y)

limits are y-coordinates from bottom to top

Volume

Method of Circular Disk Conditions

1. element should be perpendicular to axis

2. axis is a boundary

Volume - Method of Circular Disk

Vertical Element

V = π ∫y^2 dx

y = Y-above - Y-below (in terms of x)

limits are x-coordinates from left to right

Volume - Method of Circular Disk

Horizontal Element

V = π ∫x^2 dy

x = x to the right - x to the left (in terms of y)

limits are y-coordinates from bottom to top

Volume

Method of Circular Ring/Washer Conditions

1. element is perpendicular to axis

2. axis is not a boundary

Volume - Method of Circular Ring/Washer

Vertical Element

V = π ∫(R^2 - r^2)dx

R = Y-above - Y-below (with bigger distance from axis) r = Y-above - Y-below (with smaller distance from axis) in terms of x

limits are x-coordinates from left to right

Volume - Method of Circular Ring/Washer

Horizontal Element

V = π ∫(R^2 - r^2)dy

R = x to the right - x to the left (with bigger distance from axis) r = x to the right - x to the left (with smaller distance from axis) in terms of y

limits are y-coordinates from bottom to top

Volume

Method of Cylindrical Shell Conditions

1. element is parallel to axis

2. axis may/may not be a boundary

Volume - Method of Cylindrical Shell

Vertical Element

V = 2π ∫xydx

x = horizontal distance between element and axis y = vertical element between element and axis (in terms of x)

limits are x-coordinates from left to right

Volume - Method of Cylindrical Shell

Horizontal Element

V = 2π ∫xydy

x = horizontal distance between element and axis y = vertical element between element and axis (in terms of y)

limits are y-coordinates from bottom to top

ρ (weight density of the fluid) values

62.4 lbs/ft^3

9800 N/m^3

Hydrostatic Pressure

F = ρ ∫hdA

ρ = weight density of fluid h = distance between element and surface of liquid dA = differential area (in terms of y) = (Xr - Xl)dy

limits are y-coordinates from bottom to top

Work done in pumping the content of a tank

W = ρ ∫hdV

ρ = weight density of fluid h = distance between element and outlet dV = differential volume (in terms of y) = method of circular disk/circular ring or washer/cylindrical shell

limits are y-coordinates from bottom to top

Work done in stretching/compressing a spring

W = k ∫xdx

always k [ 1/2 x^2] (tip: formula for work due to spring from p6a)

k = spring constant (N/m) x = displacement

limits are from initial position to final position

Work done in lifting a leaking bucket

W = ∫F(x)dx

note: always initial minus final

getting points of intersection with x and y axis

at x-axis

slope

rise/run

y2-y1/x2-x1

equation of line (slope-point form)

y - y1 = m(x - x1)

equation of circle

( x - h )^2 + ( y - k )^2 = r^2

equation of parabola

y = a(x-h)^2 + k

x = a(y-k)^2 +h

standard equation of a regular parabola

y^2 = 4ax

x^2 = 4ay

equation of ellipse

x^2/b^2 + y^2/a^2 = 1

x^2/a^2 + y^2/b^2 = 1

equation of hyperbola (sideways)

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

equation of hyperbola (opens up and down)

(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1