HCH2: Integration Flashcards
Definite integral of f(x)
∫ b f (x) dx
a
Area under a curve
Integral of f(x)
∫ b f (x) dx
a
Indefinite integral of f(x)
∫ f(x) dx = f(x) + C
Where ∫ (ax + b)^n
∫ (ax + b)^n dx = (ax+b)^n-1
—————————
a (n+1)
What are the two approximation methods for finding the area under a curve?
- Simpsons
- Trapezoidal
Simpsons Rule
A ≈ h [ dF + 4dM + dL ]
–
3
where h= (b - a) ÷ 2
Each application requires three function values
For more than one application dL1 = dF2
No subunits = no function values + 1
no function values = no subunits - 1
Trapezoidal Rule
A ≈ h [ f (a) + 2 f(b) + f (l)
———
2
where h = (b - a) ÷ n
Areas below the x-axis
A = | ∫ b f(x) dx |
a
Areas that cut between axis
A = | ∫ b f (x) dx | + ∫ c f (x) dx
a b
Areas between two curves
A = ∫ b [ f (x) - g (x) ] dx
a
you may need to plus the two areas together
Areas bounded by the y-axis
Rearrange equation to make y the subject
Limits b and a will be on y axis
Volume of a curve rotated about the x axis
V = π ∫ b y^2 dx
a
Volume of a curve rotated about the y axis
V = π ∫ b x^2 dy
a
