Integration Flashcards
Integration vocabulary
antiderivative notation ∫ symbol f(x) symbol dx anti-differentiation process integrate
The antiderivative of f is denoted by ∫ f(x) dx
The symbol ∫ is called an integral sign
f(x) is called an integrand
The dx specifies that this is the integral of f(x) with respect to x
The process of anti-differentiation can also be called integration
To integrate is to antidifferentiate, the result of integration is called the integral or antiderivative
Indefinite integral
meaning
example
issue
note
This type of integral is called an indefinite integral because the value of c is unknown
If f(x) = x² is differentiated then the result is f'(x) = 2x Then the antiderivative of f'(x) = 2x will be, ∫ 2x dx = x² + c
The issue with this is that many other functions differentiate to 2x,
f(x) = x² + 6 or f(x) = x² - 20 etc
To allow for this possibility a constant of integration c or arbitrary constant has to be added each time we integrate
The indefinite integral is a function and the definite integral is a number
Power rule
meaning
formula
To antidifferentiate a power of x, increase the exponent by 1 and the divide by the new exponent
∫ axⁿ dx = [axⁿ⁺¹ / n + 1] + c where n ≠ -1
Properties of indefinite integrals (3)
∫[ f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
∫[ f(x) - g(x)] dx = ∫ f(x) dx - ∫ g(x) dx
∫ kf(x) dx = k x ∫ f(x) dx , where k is a constant
Finding the constant of integration
If enough information is given (gradient, coordinates etc) then the value of the constant c may be found.
Integration chain rule
∫ k(ax + b)ⁿ dx
{ [ k(ax + b)ⁿ⁺¹ ] / a(n + 1) } + c
Further integration process
Given that y = 48x(3x² - 4)⁷
and asked to find:
∫ 6x(3x² - 4)⁷ dx
∫ 6x(3x² - 4)⁷ dx = ∫ {48x/6} (3x² - 4)⁷ dx
= {1/8} ∫ 48x(3x² - 4)⁷ dx
= {1/8} (** integral of 48x(3x² - 4)⁷)
= {1/8} (3x² - 4)⁸ + c
Definite Integration
uses
evaluating
calculation
Uses of integration can include finding the area between a curve and the x-axis, can be found evaluating a definite integral.
Evaluating a definite integral gives a number
A definite integral is calculated by integrating a function between 2 values, called the limits of integration, b(upper boundary and a(lower boundary) b > a, these 2 values are substituted into the integrated function and the difference is taken
Definite integral formula
∫ ᵇₐ f’ (x) dx = [f(x)ᵇₐ] = f(b) - f(a) where b > a
constant c is not shown as it is cancelled out
Properties of definite integral
∫ᵇₐ f(x) dx = - ∫ᵃᵦ f(x) dx where b > a
∫ ᵃₐ f(x) dx = 0
∫ᵇₐ f(x) dx + ∫ᶜₐ f(x) dx
Area enclosed by a curve and the x-axis
uses
case/scenario
formula
Differentiation gives the gradient of a curve at a particular point, integration can be used to find volumes, heat and mass too
If y = f(x) is a function where y ≥ 0 then the area A bounded by the curve is given by the formula
∫ᵇₐ f(x) dx where b > a
Area above and below the x-axis
uses
formula
process
Sometimes when the curve crosses the x-axis part of the area may be above the x-axis and part below the x-axis, to evaluate this treat each area separately and are regarded as positive, then added
∫ᵇₐ f(x) dx + | ∫ᶜᵦ f(x) dx | where c > b > a
To find the boundaries we factorise the function and then to calculate the area we integrate the function at the boundaries, max and min stationary points may also be found including nature to help sketch the graph to visualise the areas
Area enclosed by a curve and the y-axis
If x = f(y) is a function with x ≥ 0, then the area A bounded by the curve is given by the formula
∫ᵇₐ f(y) dy where b > a
Area bounded by a curve and a line or by two curves
A = ∫ᵇₐ [top function - bottom function] dx where b > a
A = ∫ᵇₐ [top function] dx - ∫ᵇₐ [bottom function] dx where
b > a
Note: if the graphs intersect then we need to be aware of signed areas
Improper integral meaning
Where at least one of the limits is ± ∞
Where the function to be integrated is not defined at a point in the interval of integration
