Integral Calculus

Indefinite Integrals

Definition: A function F(x) is the antiderivative of a function ƒ(x) if for all x in the domain of ƒ,

ƒ(x) dx = F(x) + C, where C is a constant.

Basic Integration Formulas

General and Logarithmic Integrals

1. kƒ(x) dx = k ƒ(x) dx	2. [ƒ(x)g(x)] dx = ƒ(x) dxg(x) dx
3. k dx = kx + C	4. xn dx = + C, n-1
5. ex dx = ex + C	6. ax dx = + C, a 0, a1
7. = ln |x| + C

Trigonometric Integrals

1. sin x dx = -cos x + C	2. cos x dx = sin x + C
3. sec2 x dx = tan x + C	4. csc2 x dx = -cot x + C
5. sec x tan x dx = sec x + C	6. csc x cot x dx = -csc x + C
7. tan x dx = -ln |cos x| + C	8. cot x dx = ln |sin x| + C
9. sec x dx = ln |sec x + tan x| + C	10. csc x dx = -ln |csc x + cot x| + C
11.	12.
13.

Integration by Substitution

ƒ(g(x))g'(x) dx = F(g(x)) + C

If u = g(x), then du = g'(x) dx and ƒ(u) du = F(u) + C

Integration by Parts

u dv = uv - v du

Distance, Velocity, and Acceleration (on Earth)

a(t) = s''(t) = -32 ft/sec2

v(t) = s'(t) = s''(t) dt = -32 dt = -32t + C1

at t = 0, v0 = v(0) = (-32)(0) + C1 = C1

s(t) = v(t) dt = (-32t + v0) dt = -16t2 + v0t + C2

Separable Differential Equations

It is sometimes possible to separate variables and write a differential equation in the form

ƒ(y) dy + g(x) dx = 0 by integrating:

ƒ(y) dy + g(x) dx = C

Exercise:	Solve for
	2x dx + y dy = 0
	x2 + = C

Applications to Growth and Decay

Often, the rate of change or a variable y is proportional to the variable itself.

= ky		separate the variables
= k dt		integrate both sides
ln |y| = kt + C1
y = Cekt	Law of Exponential Growth and Decay
	Exponential growth when k 0
	Exponential decay when k 0

Definition of the Definite Integral

The definite integral is the limit of the Riemann sum of ƒ on the interval [a, b]

ƒ(x) dx

Properties of Definite Integrals

1. [ƒ(x) + g(x)] dx = ƒ(x) dx + g(x) dx

2. kƒ(x) dx + kƒ(x) dx

3. ƒ(x) dx = 0

4. ƒ(x) dx = -ƒ(x) dx

5. ƒ(x) dx + ƒ(x) dx = ƒ(x) dx

6. If ƒ(x) g(x) on [a, b], then ƒ(x) dx g(x) dx

Approximations to the Definite Integral

Riemann Sums

ƒ(x)dx = Sn =

Trapezoidal Rule

ƒ(x)dx [ƒ(x0) + ƒ(x1) + ƒ(x2) + ... + ƒ(xn-1) + ƒ(xn)]

The Fundamental Theorem of Calculus

If ƒ is continuous on [a, b] and if F' = ƒ, then

ƒ(x) dx = F(b) - F(a)

The Second Fundamental Theorem of Calculus

If ƒ is continuous on an open interval I containing a, then for every x in the interval,

ƒ(t) dt = ƒ(x)

Area Under a Curve

If	ƒ(x)0 on [a, b]	A = ƒ(x) dx
If	ƒ(x)0 on [a, b]	A = -ƒ(x) dx
If	ƒ(x)0 on [a, c] and	A = ƒ(x) dx - ƒ(x) dx
	ƒ(x)0 on [c, b]

Exercise	The area enclosed by the graphs of y = 2x2 and y = 4x + 6 is:
	(A) 76/3
	(B) 32/3
	(C) 80/3
	(D) 64/3
	(E) 68/3
The answer is D.	Intersection of graphs:	2x2 = 4x + 6
		2x2 - 4x + 6 = 0
		x = -1, 3
	A = 4x + 6 - 2x2
	= (2x2 + 6x - )
	= 18 + 18 - 18 - (2 - 6 + 2/3)
	= 64/3

Average Value of a Function on an Interval

ƒ(x) dx

Volumes of Solids with Known Cross Sections

1. For cross sections of area A(x), taken perpendicular to the x-axis:

V = A(x) dx

2. For cross sections of area A(y), taken perpendicular to the y-axis:

V = A(y) dy

Volumes of Solids of Revolution: Disk Method

V = r2 dx
Rotated about the x-axis:	V = [ƒ(x)]2 dx
Rotated about the y-axis:	V = [ƒ(y)]2 dy

Volumes of Solids of Revolution: Washer Method

V = (ro2 dx - ri2 ) dx
Rotated about the x-axis:	V = [(ƒ1(x))2 - (ƒ2(x))2] dx
Rotated about the y-axis:	V = [(ƒ1(y))2 - (ƒ2(y))2] dy

Exercise:	Find the volume of the region bounded by the y-axis, y = 4, and y = x2
	if it is rotated about the line y = 6.
	[(x2 - 6)2 - (4 - 6)2 ]dx
	= cubic units

Volumes of Solids of Revolution: Cylindrical Shell Method

V = 2rh dr
Rotated about the x-axis:	V = 2xƒ(x) dx
Rotated about the y-axis:	V = 2yƒ(y) dy