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IB Math IA- evaluating definite integrals

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Introduction

        In this assignment we are being asked to discover a way to evaluate definite integrals independently of our calculator. In order to proceed with the investigation, we must identify the meaning of a definite integral.

A definite integral is defined as the area between a curve in the given x-axis in a given interval. The definite integral of the function f from x = a to x = b gives a way to find the product of (b-a) and f(x), even if f is not a constant. The definite integral is also defined as an average rate of change and can be written as:

image00.png

Given the information above, investigate the definite integral of 3sin(2x) defined as:

I(b) = image09.pngimage01.png

← Graph of 3sin(2x)

We can begin the investigation by plotting points on a graph to represent values of the definite integral. For this trial we will keep A=0 and B will be manipulated.

image14.png

Start by plugging in the equation to the calculator:

Ex. 3sin(2x) where x= π/2

       3sin(2image24.pngπ/2)= 0

Ex. 3sin(2x) where x= π/6

       3sin(2image24.pngπ/6)= 2.60

These are the y values for the function. Then graph

the (x,y) coordinates. The area above the x=axis and

in between the two points is considered the definite integralimage36.png

or area under the curve. Next, shade the area

under the curve from 0-

...read more.

Middle

-1.5

-π/6

-1.37

-1.73

0.69

-2.25

-π/12

-1.92

-2.28

0.141

-2.80

1

0

-0.356

2.06

-0.876

2

0.356

0

2.421

-0.520

3

-2.06

-2.42

0

-2.94

4

-0.406

-0.762

1.66

-1.28

-1

0

-0.356

2.06

-0.876

-2

0.356

0

2.42

-0.520

-3

-2.06

-2.42

0

-2.94

-4

-0.406

-0.762

1.66

-1.28

2.25

-0.308

-0.664

1.76

-1.18

2.5

-1.05

-1.41

1.01

-1.93

2.75

-1.69

-2.04

0.377

-2.56

3.25

-2.09

-2.45

-0.025

-2.96

3.5

-1.76

-2.11

0.309

-2.63

-2.25

-0.308

-0.664

1.76

-1.18

-2.5

-1.05

-1.41

1.01

-1.93

-2.75

-1.69

-2.04

0.377

-2.56

-3.25

-2.09

-2.45

-0.024

-2.96

-3.5

-1.76

-2.11

0.309

-2.63


The B values stay the same but the Yimage07.pngimage08.pngimage06.png

Values will change because we manipulated

Now we must try to generate a formula for image04.png in terms of A and B. In the first steps we only made a formula in terms of b. Now we are taking into account both variables to try and reach a more general solution.

Trials:

Start by trying to substitute x for A and B.

Ex. Where A=1 and B=π/3

-3/2cos(2x)+1.5 is our general formula right now

-3/2cos(2image10.pngπ/3)+ 1.5 = 2.25

-3/2cos(2image10.png1)+ 1.5 = 2.12

image11.png(We are given this information from the table above)

Ex. Where A=1 and B= π/2

-3/2cos(2image10.pngπ/2)+ 1.5= 3

-3/2cos(2image10.png1)+ 1.5= 2.12

image12.png (We are given this information from the table above)

From these values we can try subtracting A-B to see if the value will be the same as the value from the definite integral I(b)

2.12-2.25=-1.3

2.12-3= -0.88

These values are similar to the actual value but are the opposite sign. If we subtract B-A instead of A-B, we will get the correct value.

2.25-2.12= 1.3image13.png1.26
3-2.12= 0.88
image13.png0.876

If we know that by subtracting the values of the two functions from one another then the formula for image04.png in terms of a and b is 3cos(B+ π/2)2 – 3cos(A+ π/2)2

Further Examples:

image15.png= 3cos(π/3+ π/2)2 – 3cos(2+ π/2)2= -.230

image16.png= 3cos(π/6+ π/2)2 – 3cos(3+ π/2)2= .690

image15.png= 3cos(π/12+ π/2)2 – 3cos(π/2+ π/2)2= -2.80

In order to reach a more generic function we need to investigate more than one function. For a second function we will use image18.pngimage17.png

Start by graphing the function without the integral.

To get the value of the area (definite integral)

Plug FnInt (Y1, X, A, B) into the graphing calculator:

Ex. FnInt (5-4x, x, 0, π/2)= 2.92

Ex. FnInt (5-4x, x, 0, π/6)= 2.07

B

Y (A=0)

0

0

π/2

2.92

π/3

3.04

π/4

2.69

π/6

2.07

π/12

1.17

-π/2

-12.8

-π/3

-7.43

-π/4

-5.02

-π/6

-3.17

-π/12

-1.45

1

3

2

2

3

-3

4

-12

-1

-7

-2

-18

-3

-33

-4

-52

2.25

1.13

2.5

0

2.75

-1.38

3.25

-4.88

3.5

-7

-2.25

-21.4

-2.5

-25

-2.75

-28.9

-3.25

-37.4

-3.5

-42

...read more.

Conclusion

image24.pngπ/2)+ 4(π/22))= 0.081

Ex. image30.png= ((5image24.pngπ/2)+ 4(π/22))- ((5image24.png-1)+ 4(-12))= 9.92

From this information, we can formulate a shortcut to find image00.pngfor any generic function f(x).

image00.png= image31.pngWhereimage32.pngis the symbol for the anti-derivative

In order to ensure that this shortcut is correct, we must test it out with other example functions. We can try other functions that would require us to apply some of the anti-differentiation rules such as the anti-power rule =image33.png

image34.pngThe anti-derivative of this function= 1/3x3

image35.png= ((1/3)(13)- ((1/3(03)= .333-0= .333

image37.png= ((1/3)(23)-((1/3)(13)= 2.67-.333= 2.33

image38.png= image39.png

= image40.png

= image41.png

Now substitute X for values of A and B

image42.png

image43.png

Apply the general formula where image00.png= image31.png

image43.png- image42.png

A=1 and B=2

image45.png- image46.png

image47.png- image48.png= image49.png= .25

Now plug in on the calculator:

FnInt((x3-2)/(x3), x, 1, 2

this calculation equals .25 so the generic function does work for this equation.

Assuming that the function of the definite integral itself can be anti-differentiated, then the generic formula we found image00.png= image31.pngshould work for all functions at any values of A and B. This helps us to establish a universal rule that can be applied to several different functions. Through this investigation, we proved that you could find a pattern for a mathematical concept by investigating a functions properties and applying our knowledge of calculus and other mathematical concepts to reach an answer. We applied what we already knew about definite integrals, derivatives, anti-derivatives, etc. to develop a formula that now works for all generic functions when you are trying to derive the anti-derivative that is equivalent to the value of the definite integral.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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