Indefinite Integrals
Definition: A function F(x) is the antiderivative of a function ƒ(x) if for all x in the domain of ƒ,
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
Integration by Parts
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:
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]
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
The Second Fundamental Theorem of Calculus
If ƒ is continuous on an open interval I containing a, then for every x in the interval,
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
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 |