INTEGRATION

INTEGRATION

Techniques of anti-derivatives x n+1 + c , n ≠ −1 n +1 dx ± ∫ g ( x) dx

Power rule

n ∫ x dx =

Sum and difference

∫ [ f ( x) ± g ( x)] dx = ∫ f ( x) ∫ kf ( x)
1

Constant multiple rule

dx = k ∫ f ( x) dx , k constant

Anti-derivatives of logarithmic

∫ ax dx = a ln | x | +c ; ∫ ax ∫ e dx =

1

f ′( x) dx = ln | f ( x) | + c f ( x)

Anti-derivatives of exponential

e ax + c; a

∫ f ′( x)e

f ( x)

dx = e f ( x ) + c

∫ cos x dx = sin x + c ∫ sin x dx = − cos x + c ∫ tan x dx = − ln | cos x | +c
Anti-derivatives of trigonometric

∫ cot x dx = ln | sin x | +c ∫ sec x dx = ln | sec x + tan x | +c ∫ csc x dx = − ln | csc x + cot x | +c ∫ sec ∫ csc
2

x dx = tan x + c x dx = − cot x + c

2

∫ sec x tan x dx = sec x + c ∫ csc x cot x dx = − csc x + c

1

INTEGRATION

Techniques of integrations: - substitutions

∫ f ( g ( x)) g ′( x) dx = F(g(x)) + c u = g(x), du = g′ (x) dx ⇒

∫ f (u)du = F (u ) + c

Area between two curves
If f(x) > g(x) for all x in [a, b], then the area of the region between the graphs of f(x) and g(x) and between x = a and x = b is given by

A = ∫ [ f ( x) − g ( x)] dx a b

Double integrals
(a) Computing the double integral over a rectangular If R is the rectangular a < x < b and c < y < d, then

∫∫
(b)

R

d b b d     f ( x, y ) dxdy = ∫  ∫ f ( x, y ) dx  dy = ∫  ∫ f ( x, y ) dy  dx     ca ac  

Computing the double integral over a nonrectangular region If R is the region a < x < b and c(x) < y < d(x), then

∫∫
(c)

R

b  d ( x)  f ( x, y ) dxdy = ∫  ∫ f ( x, y ) dy  dx   a  c( x) 

Computing the double integral over a nonrectangular region If R is the region a(y) < x < b(y) and c < y < d, then

∫∫

R

 b( y )  f ( x, y ) dxdy = ∫  ∫ f ( x, y ) dx  dy   c  a( y)  d 2

INTEGRATION

AREA BETWEEN TWO CURVES

1.

Find the area under the graph of f (x) = 2 over the interval [1, 4]. [6 unit2] Find the area under the graph of f (x) = 1 – x2 over the interval [0, 1]. [2/3 unit2]

2.

3.

Find the total area of the region enclosed by the graph of y = xe x , the x-axis, and the vertical lines x = −1 and x = 1. [1.71828 unit2] Find the area of the region between f (x) = − x 2 − 3 x + 4 and g ( x ) = x 2 − 3 x − 4 , and between x = −1 and x = 1. [44/3 unit2]
Find the area of the region between y = 3x2 and y = 1 – x2 and between x = 0 and x = 1. [1 unit2] Find the area enclosed by y = x2 and y = x3. [1/12 unit2] Find the area bounded by the curves y2 = 8x and y = x2. [8/3 unit2] Sketch the graphs x3y = 1 and y = x in the same diagram for positive values of x. Show that the finite area bounded by lines y = 0 and x = 2, and the arcs of each curve 25 is 24 .

2

4.

5.

6.

7.

8.

3

INTEGRATION

DOUBLE INTEGRALS If R is the rectangle 1 ≤ x ≤ 2 and 1 ≤ x ≤ 3, then compute (a) (b)

1.

∫∫ ∫∫

R

1 dxdy 1 dydx
[(a) 2, (b) 2]

R

2.

If R is the rectangle 0 ≤ x ≤ 1 and 0 ≤ x ≤ 2, then compute (a) (b)

∫∫ ∫∫

R

xy dxdy xy dydx
[(a) 1, (b) 1]

R

3.

Consider the figure.

Compute

∫∫

R

x dxdy .
[8/3]

4.

Evaluate
1 1

(a)

∫ ∫ ( x − 2 y)
0 0 1 2

dxdy
[−1/2]

(b)

∫ ∫ ( ye
0 0
1 x2

x

− x − y ) dxdy
[ 1 e2 − 3 ] 2

(c)

∫∫
0 − x2

x dydx [1/2]

4