• 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. Math IB SL Shady Areas Portfolio

    + g(0.2)], the second as [g(0.2) + g(0.4)], and so on until the 5th trapezoid as [g(0.8) + g(1)]. This gives the following calculation: Factoring out and: The area approximated by using 5 trapezoids differs in value to the approximation using only 2 trapezoids. Examine the following diagrams of the approximated area using 1-8 trapezoids: Diagram

  2. Investigating ratio of areas and volumes

    Now the conjecture must be tested for different arbitrary points. Consider the points between x = 0 and x = 2. Area A will be the area contained between the graph of y = xn and the x-axis between the points x = 0 and x= 2.

  1. 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.

  2. type 1 maths portfolio trapezium rule

    approximate the area under the curve of the function , the better approximation and the nearer you come to the area derived when calculated using the integral. With help of technology, the area under a curve approximated using the trapezoid rule requires splitting the interval into n subintervals of length based on equally spaced points with corresponding ordinates.

  1. Maths Coursework on Integration

    We have hence created a provisional formula for the approximate. This gives the following formula for 5 trapeziums The formula can then be simplified: Inserting the values for h and the different heights gives: Looking at the results for the approximate area under the curve using two and five trapeziums

  2. Investigation of Area Under a Curve and Over a Curve

    Throughout the rest of the investigation will be using the method 2 to solve the area of A. I will use the second method to find the area for the first few values of n following 2. The next equation I will find the ratio for is.

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

    (h1 + h2) =(1/2) (0.5) (3 + 3.25) = 1.5625 TA= AA + AB TA= 1.5625 + 1.8125 TA= 3.375 Increase the number of trapeziums to five and find a second approximation for the area. In the next part, the number of trapeziums has increased to five and the goal is to find a second approximation for the area.

  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