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1

Integrals and anti-derivatives topics

  1. Riemann Sums
  2. Definite integrals
  3. Fundamental Theorem of Calculus

2

A Riemann Sum equivalent to the definite integral from a to b of f(x)dx

limit as delta x -> 0 of the sum of f(xi)deltax for a

a Riemann sum is an approximation:

 

 

3

Given the anti derivative of any given function, what will you almost always need to add at the end?

C, the constant that will go to 0 once differentiated

4

integral from a to b f(x)dx

f(b) - f(a) where F' = f

5

Derivative with respect to x of an integral from a to x? ( f(t) and dt )

f(x)

6

d/dx of a to g(x) f(t)dt?

f(g(x))*g'(x)

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Given the fact that an antiderivative is the opposite direction of the norm, what must we consider?

We do not have the same level of certainty: the constant, C, accounts for this.

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SS signifys antiderivative: SS sec u * tan u du

 

sec u + C

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SS csc u * cot u du

-csc u + C

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SS tan u du

ln |sec u| +C

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What is the antiderivative of cot(x)?

1. cot(x)=cos(x)/sin(x)

2. Let t=sin(x);

=> dt=cos(x)dx.

3. Integral 1/t == ln |t|

=> Integral cot(x) = ln | sin(x) | + C

ln(|x|)+C

12

Integral(square(csc(x)))

-cot(x) + C

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SS secu du

ln| secu + tanu | + C

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SS cscu du

-ln| csc u + cot u | +c

15

SS du(sqrt (1-u2) )

arcsin u + C

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SS du/(1+u2)

arctanu + C

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SS du/(u*sqrt(u2 -1))

arcsec u u du

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SS eu du

eu + C

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SS au du

 au/ln a

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SS du/u

ln |u| + C

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Displacement

Integral of velocity with respect to time 

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Distance

Absolute value of the integral of velocity with respect to time:

 

distance is different from displacement. Cannot have negative distance.

23

Area between f and g, where f(x) >= g(x) from a to b

SS from a to b of ( f(x) - g(x) ) dx

24

For any problem dealing with change:

 

If R(t) represents a rate of change, then the SS fomr a to b R(t)dt represents...

the accumulated amount of change during the timer interval from a to b.

 

MPH ==> total miles driven (minus negative displacement)

10 gals/sec ==> how many gallons are in the pool

etc etc

25

Volume by disks

1. Rotate function about x-axis- dx rectangles become dx-height cylinders e.g "disks"

2. Same approximating model- limit as dx goes to 0 of sum of disks

3. Volume of each disk ==> pi*r2*h = pi * ( f(x) )2 * dx

4. SS pi*f(x)2 dx

26

Volume by washers

dV = pi(R2 - r2) dx

27

How do you compute the  Left Riemann Sum of a function?

For the left Riemann sum, approximating the function by its value at the left-end point gives multiple rectangles with base Δx and height f(a + iΔx). Doing this for i = 0, 1, ..., n − 1, and adding up the resulting areas gives: 

 

28

The left Riemann sum amounts to an overestimation if f is ?

The left Riemann sum amounts to an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.

29

What is the Left Riemann Sum of a function? 

 

  • Let 
  • If  for all i, then S is called a left Riemann sum.

30

Draw a diagram of the  Left Riemann sum  of x3over [0,2] using 4 subdivisions

 

 Left Riemann sum  of x3over [0,2] using 4 subdivisions
 

 

31

Draw a diagram of the right Riemann sum methods of x3over [0,2] using 4 subdivisions

 
Riemann sum methods of x3over [0,2] using 4 subdivisions
 
Right
 

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Riemann sum methods of x3over [0,2] using middle subdivisions

USE MRAM

33

Draw a diagram of Riemann sum methods of x3over [0,2] using trapezoidal subdivisions.

Riemann sum methods of x3over [0,2] using trapezoidal subdivisions

34

Compute the area under the curve of y = x2 between 0 and 2  using Riemann's method.

  1. The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of ; these are the widths of the Riemann rectangles (hereafter "boxes").
  2. Because the right Riemann sum is to be used, the sequence of x coordinates for the boxes will be .
  3. Therefore, the sequence of the heights of the boxes will be .
  4. It is an important fact that , and .
  5. The area of each box will be  and therefore the nth right Riemann sum will be:
  1.  Take the limit as n → ∞

This method agrees with the definite integral as calculated in more mechanical ways:

35

Draw the diagrams for the 4 Riemann Sum methods for an arbitrary function f.

 

Four of the Riemann summation methods for approximating the area under curves.

  1.  Right and leftmethods make the approximation using the right and left endpoints of each subinterval, respectively.
  2. Maximum and minimum methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively.
  3. The values of the sums converge as the subintervals halve from top-left to bottom-right.

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