Integral Calculus Formula Sheet Page 3
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1
2
3
4Areas and Volumes:
Area in terms of x (vertical rectangles):
Area in terms of y (horizontal rectangles):
b
d
(
top bottom dx
)
(
right left dy
)
a
c
General Volumes by Slicing:
Disk Method:
Given: Base and shape of Cross‐sections
For volumes of revolution laying on the axis with
slices perpendicular to the axis
b
V
A x dx
( )
if slices are vertical
b
2
if slices are vertical
V
R x
( )
dx
a
d
a
if slices are horizontal
V
A y dy
( )
d
2
V
R y
( )
dy
if slices are horizontal
c
c
Washer Method:
Shell Method:
For volumes of revolution not laying on the axis with
For volumes of revolution with slices parallel to the
slices perpendicular to the axis
axis
b
b
2
2
if slices are vertical
if slices are vertical
V
R x
( )
r x
( )
dx
V
2
rhdx
a
a
d
d
2
2
V
R y
( )
r y
( )
dy
if slices are horizontal
V
2
rhdy
if slices are horizontal
c
c
Physical Applications:
Physics Formulas
Associated Calculus Problems
Mass:
Mass of a one‐dimensional object with variable linear
Mass = Density * Volume (for 3‐D objects)
density:
Mass = Density * Area (for 2‐D objects)
b
b
Mass
(
linear density dx
)
( )
x dx
Mass = Density * Length (for 1‐D objects)
distance
a
a
Work:
Work to stretch or compress a spring (force varies):
Work = Force * Distance
b
b
b
Work
(
force dx
)
F x dx
( )
kx
dx
Work = Mass * Gravity * Distance
Hooke s Law
'
Work = Volume * Density * Gravity * Distance
a
a
a
for springs
Work to lift liquid:
d
Work
(
gravity density distance area of a slice dy
)(
)(
) (
)
c
volume
d
W
9.8* * ( ) *(
A y
a y dy
)
(
in metric
)
c
Force/Pressure:
Force of water pressure on a vertical surface:
Force = Pressure * Area
d
Force
(
gravity density depth width dy
)(
)(
) (
)
Pressure = Density * Gravity * Depth
c
area
d
F
9.8* *(
a y
) * ( )
w y dy
(
in metric
)
c
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