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Integral Calculus
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Term
Area between curves
Integral Calculus
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Definition
To determine the area
bounded by two curves:
Integral Calculus
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An example problem…
Determine the area bounded by the
curves y = x2 and y = x3
Intersection point
Decide dx or dy
Integral set-up
Organize your terms
Take the integral
Area under a curve
To determine the area under a
curve,
Graph it
Limits
Use the integral
Evaluate
Differential Equations
To solve a differential equation
Separate the dy from the dx
Use an integral on both sides
Check any initial condition
Solve for y
Determine the area bounded by
y = 4x², x = 1 and x = 3
Solve the differential equation
dy/
dx
= 4x²y
Disc method [volume]
Washer Method
To determine the volume of a
solid using the disc method:
Pi goes out front
Limits
Outer2 – Inner2
Work it, Baby
Indefinite integrals
Indefinite integrals are not
evaluated using any limits
Determine the volume of the solid
generated by rotating the region
bounded by y = x2, x = 2 and x = 8
around the x-axis
Integrate the following:
 3x
4
 3x
2
+ 2x dx
– 5x dx
is a prime example. Indefinite
integrals and antiderivatives
are somewhat synonymous –
although an antiderivative is
the “undoing” of a derivative
which may or may not be
evaluated using given limits
 [sin x]
dx
Since the derivative of any
constant is zero, a “+ C” is
included in the solution in
order to include all possible
solutions which may exist
Integrating 1/x

1/
x
dx = ln x + C
so anytime a function is in that
form, your integral will result in
some type of natural log
Since negative numbers can’t
yield natural log values, the
absolute value sign is used
Determine the area bounded by
y = 1/x, x = 1 and x = 10
Integrating
polynomial functions
When integrating polynomial
functions, the power goes up
by one and its coefficient
becomes the reciprocal of the
resulting power
x
Integrating trig
functions
4
x
3
– 4x2 + 12x + 3 dx
dx = 1/5 x5 + C
When integrating trigonometric
functions remember
Integrate the following:
 [sin x] dx =
-cos x + C
 sin 2x
dx
 [cos x] dx =
sin x + C
 [sec x] dx =
tan x + C
 x sec
[x2] dx
2
 [sec x tan x] dx =
L'Hopital's Rule
Integrate the polynomial function
sec x + C
To determine the limit of a
0
function in the form of /0,
Determine the limit of

/ or any other indeterminate
form, simply take the
derivative of the top AND of
the bottom and re-evaluate the
function
2
lim 0
cos [2x] – 1
sin x
Shells method
[volume]
To determine the volume of a
solid using the shell method
2 goes out front
Put an x next
Limits
A function [or difference] goes
next
Calculate the volume
Trigonometric
substitution
In order to integrate functions
in the forms of
 a2 – x2
 x2 – a2
 a2 + x2
use the following steps
Bust the right triangle
Get x
Get dx
Get the triangle rebuilt
Get it integrated
u substitutions
In order to use u substitutions:
Substitute as u
Hot Mama
Adjust your limits
Function switch
Take the integral
Determine the volume of the solid
generated by revolving the regions
bounded by y = x3, y = 1 and y = 4
around the y-axis