Post on 21-Jan-2016
Integral calculation. Indefinite integral
Differentiation Rules
If f(x) = x6 +4x2 – 18x + 90
f’(x) = 6x5 + 8x – 18
*multiply by the power, than subtract one from the power.
Chain rule in transcendental
Take for example
y=esin(x)….let sin(x) = u gives; du/dx = cos(x)
y=eu….dy/du =eu
Use chain rule formula
dy/dx= eu.cos(x)
= cos(x).esin(x)
Thus, assign the function that is inside of another function “u”, in this case sin(x) in inside the exponential.
Integration
Anti-differentiation is known as integration
The general indefinite formula is shown below,
Integration
FORMULAS FOR INTEGRATIONGENERAL Formulae
Trigonometric Formulae
Exponential and Logarithmic Formulae
Linear bracket Formula
Indefinite integrals Examples
• ∫ x5 + 3x2 dx = x6/6 + x3 + c
• ∫ 2sin (x/3) dx = 2 ∫ sin(x/3) dx = -2x3cos(x/3) + c
• ∫ x-2 dx = -x-1 + c
• ∫ e2x dx = ½ e2x + c
• ∫ 20 dx = 20x + c
Definite integrals
1 3 x
yy = x2 – 2x + 5 Area under curve = A
A = ∫1 (x2-2x+5) dx = [x3/3 – x2 + 5x]1
= (15) – (4 1/3) = 10 2/3 units2
3
3
Area under curves – signed area
Area Between 2 curves
Area Between two curves is found by subtracting the Area of the upper curve by Area of the lower curve.
This can be simplified into
Area = ∫ (upper curve – lower curve) dx
A = ∫-5 25-x2-(x2-25) dx
OR
A = 2 ∫0 25-x2-(x2-25) dx
OR
A = 4 ∫0 25-x2 dx
A = 83 1/3 units2
y = x2 -25
y = 25 - x2
5
5
5
Area Between 2 curves continued…
If 2 curves pass through eachother multiple times than you must split up the integrands.
y1y2
C D
A = ∫C(y1-y2)dx + ∫0(y1-y2)dx D0A1 A2
Let A be total bounded by the curves y1 and y2 area,
thus;
A = A1 +A2
Integration – Area Approximation
The area under a curve can be estimated by dividing the area into rectangles.
Two types of which is the Left endpoint and right endpoint approximations.
The average of the left and right end point methods gives the trapezoidal estimate.
y
y = x2 – 2x + 5
x
y = x2 – 2x + 5
x
LEFT
RIGHT