Unit 6 - Integration & Accumulation of Change Flashcards
Area under the curve
Area between a function and the x-axis
If we have a RoC function, we can examine its graph and determine its
accumulation of change
Units for accumulation of change is determined by
dependent units by independent unitsl
Area under the x-axis is
negative accumulation of change
Riemann Sum
Approximation method by Bernhardt Riemann used to estimate accumulation of change
The more rectange there are,
the better approximation
How to calculate the width of rectangles
b - a / n where
a= lower limit
b= upper limit
n= number of rectangles
Left Riemann Sum
Left endpoints of each rectangles touches the function
width [combined height] = approximation
height determined by plugging value of left side into function
Right Riemann Sum
Right endpoints of each rectangles touches the function
width [combined height] = approximation
height determined by plugging value of right side into function
Midpoint Riemann Sum
Midpoints of each rectangles touches the function
width [combined height] = approximation
height determined by plugging value of midpoint into function
Trapezoidal Riemann Sum
Left endpoints of each rectangles touches the function
width [sum of (left value+right value/2)] = approximation
height determined by plugging values of left value+right value into function and dividing by 2
In an increasing function, is left Riemann Sum an over or underestimate
under
In an increasing function, is right Riemann Sum an over or underestimate
over
In a decreasing function, is left Riemann Sum an over or underestimate
over
In a decreasing function, is right Riemann Sum an over or underestimate
under
In a function that is concave up, is trapezoidal Riemann Sum an over or underestimate
over
In a function that is concave down, is trapezoidal Riemann Sum an over or underestimate
under
Riemann sum can also be used to approximate in a
table of value
If n is equal to the number of sub-intervals on interval [a,b] what is the width of each sub-interval?
Delta x = b-a / n
If n approaches infinity on interval [a,b] what does the width of each sub-interval approach?
delta x =0
The sum of the area of all rectangles gives you
area under the curve
Sum of area of all rectangles can be written as
[Delta x1 * f(x1)] + [Delta x2 * f(x2)] + … + [Delta xn * f(xn)]
Definite Integral Notation
lim k=1 sum n (b-a/n) * f(a + b-a/n k)
n->oo
n = # of sub-intervals in interval [a,b]
k = kth sub-interval
This is the integral from a to b
