Integration (P1.13 & 2.11) Flashcards
∫ x^n dx =
x^n+1 / n+1 + c
indefinite vs definite integrals
indefinite: + c - produces a function
definite: integral b/w 2 limits - produces a value
describe how to find the constant of integration, c
integrate the function
substitute the values (x,y) of a point on the curve into the integrated function
solve the equation to find c
describe how to find the definite integral
[f(x)]ba = f(b) - f(a)
describe how to find area under curve
definite integral
for areas under the x-axis,
integral gives negative answer
areas b/w curves & lines
area under curve - area under line (trapezia or triangles)
practice
∫ 1/x dx =
ln|x| + c
∫ e^kx dx =
1/k e^kx + c
∫ cosx dx =
sinx + c
∫ sinx dx =
-cosx + c
describe integration by substitution
substitute a value for u
du/dx =
dx =
cancel out any x’s left
integrate
sub value back into u
∫ f’(ax+b) dx =
1/a f(ax+b) + c
reverse chain rule
only works for single powers of x
∫ k f’(x)/f(x) dx =
ln|f(x)| & adjust any constant
if integrating a fraction with no trig., look for bottom differentiating to top –> ln
tips for integration by parts
∫ uv’ dx = uv - ∫ vu’ dx
u should be a function which becomes simpler after differentiation
it must be possible to integrate v’ to find v
always make lnx term = u
sometimes you need to use integration by parts twice
if vu’ is a product of functions, you have used the wrong u & v (or need to do it twice)
practice integration using trig. identities
integrate expressions in the form kf’(x)(f(x))^n
f(x))n + 1 & adjust constant
practice integration with partial fractions
describe Y2 finding areas
the area bounded by 2 curves can be found using integration
R = ∫ba f(x) -g(x) dx
= ∫bc f(x) dx - ∫ba g(x) dx
make sure area under x-axis is made positive before finding total area (to ensure the values for area don’t cancel out)
practice exam-style integration questions with transformations
see OneNote
tips for trapezium rule
n = # of strips/trapezia
for a convex curve, trapezium rule gives an overestimate for the area
for a concave area, trapezium rule gives an underestimate for the area
explain the value for h in the trapezium rule
h = b-a / n
state # of trapezia & each has a height of h
# trapezia x h = b-a
explain why trapezium rule is an overestimate of the area under the curve
curve is convex
so lines connecting the 2 endpoints would be above the curve
so it is an overestimate
explain why trapezium rule is an underestimate of the area under the curve
curve is concave
so lines connecting the 2 endpoints would be below the curve
so it is an underestimate
how is the accuracy of the trapezium rule be improved?
more strips used
decreased width of each strip as more values are used
practice trapezium rule
describe how to solve differential equations
form differential equation: dy/dx =
separation of variables: put all y terms on the LHS & all x terms & #s on the RHS
see OneNote
how do you find the particular solution to a first-order differential equation?
if you know one point on the curve
= boundary condition
practice
modelling with differential equations
describe integration as the limit of a sum
∫ba f(x) dx = limδ->0 Σb,x=a f(x) δx
you can approximate the area under a curve as a # of thin rectangular strips
as the strips get thinner, the approximation becomes more accurate
limit notation formalises the idea of a definite integral as the limit of a sum of areas of these rectangular strips
how transformations affect area under curve
pg 329 P2 textbook
