IB Math IA- evaluating definite integrals

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        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:

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

 I(b) =    

                                ← 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.

Start by plugging in the equation to the calculator:

Ex. 3sin(2x) where x= π/2

       3sin(2π/2)= 0

Ex. 3sin(2x) where x= π/6

       3sin(2π/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 integral

or area under the curve. Next, shade the area

under the curve from 0- π/2 which represents:

There is another example of a shaded region for

the b value π/6. Through this investigation we are

trying to estimate these areas under the curve for

given values to help lead us to a formula that

will work for all values and all functions.

Instead of counting squares, using the trapezoidal method, or the midpoint method, we can estimate the value of the area/definite integral on the calculator.

To get the value of the area (definite integral)

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

Ex. FnInt (3sin(2x), x, 0, π/2)= 3

After determining one of the definite integrals for a value of b, we must continue this process to find several points for different values of b to try to find a pattern. (Only some points are shown in the table, more points are included on the graph to get a better understanding).

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All the points are now combined to create a single graph. We can start looking for patterns by observing the values for the graph of I(b) over the interval

0< b < 5 (approximate values)

0 < b < 1.6 & 3.2 < b < 4.8 the graph is increasing

1.6 < b < 3.2 & 4.8 < b < 5 the graph is decreasing

0.8 < b < 1.6 & 2.4 < b < 3.2 & 4 < b < 4.8 the graph is increasing at a decreasing rate

0 < b < 0.8 & 1.6 < b < ...

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