Unit 5: The Definite Integral

how to distinguish between an integral and derivative?

derivatives are signified by a rate of change. integrals find the total amount of change

what does the integral represent:

the area under the function (the total of what is being measured)

what does "n" represent?

the number of intervals on the x axis

what is happening if a velocity graph has positive slope?

the object is accelerating

what is happening if a velocity graph has negative slope?

the object is decelerating

what is happening if a velocity graph is concave down?

the object's speed is decreasing

what is happening if a velocity graph is concave up?

the object's speed is increasing

how to determine the units of the integral:

the units of the integral will always match those of f(x)

what must you do if there is a unit mismatch?

-use unit analysis to match units to the ones that the x values are measured in. EX: if speed is measured in km/hour and time is given in seconds, convert hours to seconds -multiple integral by this conversion

how to use the fundamental theorem of calculus:

1. find derivative of f(x) 2. show that the integral of the derivative is the same as plugging the interval endpoints into the original f(x) 3. evaluate using f(x) and endpoints

how to estimate the integral when given a table of values:

1. find the LHS and the RHS 2. average them

relationship between velocity and acceleration:

1. the derivative of velocity is acceleration 2. so the integral of acceleration is velocity

2 big ideas of the definite integral:

1. the integral of the sum/diff. if the sum/diff. of the integrals 2. if the function is a constant (ex: 2), that will be considered the height. find the integral by multiplying constant by the interval (base x height)

how to find the average integral:

1/b-a(the integral)

integral of sin(x) from 0 to pi:

2

displacement vs. distance:

displacement is only the amount of distance between the starting point and the ending point. distance is the total distance traveled.

how to find total distance when given velocity?

find the integral

the fundamental theorem of calculus:

if f is continuous on [a, b] and f(t)=F'(t) then... 1. the integral of the function is equal to the integral of the derivative function 2. integral of the derivative function is the same as finding the change in values using the original function ex: the integral of a velocity function is position because velocity is the derivative of position

how does the integral change if the interval is DESCENDING:

if its descending, that means the intervals are heading negative. the integral will then be negative

3 PROPERTIES OF DEFINITE INTEGRALS

know these. will be written on separate flashcard

how to find the left hand sum?

multiply the sum of the left hand y values by the change in t

how to find the right hand sum?

multiply the sum of the right hand y values by the change in t

what is happening if a velocity graph crosses the x axis?

object changes direction

inputs/outputs of odd functions:

opposite inputs, opposite outputs (ex: x^3)

inputs/outputs of even functions:

opposite inputs, same outputs (ex: parabola)

what does the integral of a rate of change represent?

represents the total change in f(x) between a given interval of time