• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Shady Areas

Extracts from this document...


Shady Areas

I shall investigate different ways of finding a working rule to approximate the area under a curve using trapeziums.  The area under the curve represents area between the f(x) and x values under the curve in the specified area.  Therefore, through integration to find the area, integration of the integrated area will find the volume.

                        Consider the function   g(x) = x² + 3

g(x) = x² + 3  (see Graph 1)

The area under this curve from x=0 to x=1 is approximated by the sum of the area of two trapeziums.

The approximation can be discovered by working out the area of the square in the trapezium and then by working out the triangle in the trapezium.

...read more.



=            x  ½

=            x

Y1 =                    When n = 8 using the general formulaimage05.png






                             = 1/8 x 15.8394

                             = 1.9799image44.pngimage11.png


Y2  =


                   =  1/8  x  [y0 + 2y1 +2y2 + 2y3 + 2y4 + 2y5 + 2y6 + 2y7 + y8]





Y3 =



Integration of these functions to find the

...read more.


160;           = (9x/√(x³ + 9)) =  ?



Y3 =  m        (4x³ - 23x² + 40x - 18)


     =    x^4 – (23/3)x³ + 20x² - 18x

=  (81 – 207 + 180 – 54) - (1 - (23/3) +20 – 18)

     = 0 – (-4.667)

     = 4.67




              (9x/√(x³ + 9))


              (4x³ - 23x² + 40x - 18)image37.png

The first approximation was 15.6% off from the integrated answer while the third approximation was 7.7% off from the integrated answer.  The third would be closer as the curve has a minima and maxima and, as the trapeziums overestimate with maxima and underestimates with curves with a minima, and so reduces the percentage error as it cancels it out.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Maths Portfolio Shady Areas

    to separate the X axis into, which is deciding upon how much height each section will have. Then finding the trapeziums area through doing the . Therefore for example when we are trying to find the area with 5 trapeziums the formula will be: But when we have this equation

  2. Investigating the Ratios of Areas and Volumes around a Curve

    Because occurs, we cannot have n=-1 (for the moment). Graphically it can be seen that A would always equal 0, so the rule would still hold. As we cancel the terms in the final stage, a must not equal b.

  1. Investigating ratio of areas and volumes

    y = xn between the points x = 0 and x = 2: Above the conjecture ratio area A: area B = n: 1 has been made for the graph y = xn between the points x = 0 and x = 1.

  2. The aim of this particular portfolio investigation is to find the area under a ...

    from the above it can be derived that each x value is the sum of the first interval which is zero in this case and the value of height times r where r=0,1,2,3........n. Then the area of each trapezium will be- 1/2 (g(x0)

  1. Math IB SL Shady Areas Portfolio

    examine the calculation when the same method is attempted with 5 trapezoids: Figure 3 - Five trapezoids mapped onto The height will work out to: Thus, the trapezoids will be mapped between and with equivalent heights of 0.2. Using the formula for area of a trapezoid, the total area is

  2. Shady Areas. In this investigation you will attempt to find a rule to approximate ...

    The same techniques and equations will be used in the calculations. It is predicted that since there are more trapeziums, the predicted area will be closer to the actual area (more precise). However, since there are more trapeziums, the equation for the total area is now: TA= AA + AB

  1. Approximations of areas The following graph is a curve, the area of this ...

    A = A(1)+A(2)+A(3)+A(4)+A(5) A = .604+.62+.652+.7+.764 A = 3.34 The approximated area of the graph using 5 trapeziums is 3.34 units2 Now using technology I will create two more graphs with an increasing number of trapezoids to find a pattern between the approximations of the areas so far.

  2. The investigation given asks for the attempt in finding a rule which allows us ...

    Additional trapezoids make the approximated area more accurate as the values decreases slowly, although very minimal. Possibly due to the fact that the graph is a positive curve, hence trapezoid does not extend much past the curve. Furthermore, the distance between each trapezoid can be determined via the formula: position

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work