MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 3 - NUMERICAL INTEGRATION METHODS

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1 MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL - NUMERICAL INTEGRATION METHODS This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning by example. The approach is practical rather than purely mathematical. On completion of this tutorial you should be able to do the following. Revise basic integration. Find the areas under graphs of known functions. Define ordinates and mid-ordinates. Use the mid-ordinate rule. Use the trapezoidal rule. Use Simpson's Rule Apply the methods to real engineering problems.

2 . REISION OF INTEGRATION NUMERICAL METHODS FOR INTEGRATION Lets consider how to find the area under the graph of y = f(x) = + x. A graph of this function looks like this. Figure If we solve the area by use of calculus (see tutorial ) the area would be precisely solved as follows. Over the range x = to x = the area is expressed as follows. A x x ( x )dx Carrying out the integration gives the following. x A ( x x Evaluating between limits we get the following. x A ( x x A x )dx x.. units x )dx x x x This is a precise answer and we will compare it with the results found in the following work.

3 . GRAPHICAL METHODS Consider the same function again and this time more grid lines are shown.. COUNTING RECTANGLES Figure A simple but crude way to find the area under the graph is to count the rectangles. Each rectangle on the graph above has an area of unit. Count them up judging the divided ones to the nearest half. You should get an answer of about units depending on hw good you are at ding it.. MID-ORDINATE RULE The values of y corresponding to x =, x =, x = and so on are called the ordinates. The values of y corresponding to x =.5, x =.5, x =.5 and so on are called the mid-ordinates. Figure Each column is approximately a rectangle w wide and h high. The area is approximately w h. The area under the whole graph is approximately A = w h + w h + w h +w h A = w(h + h + h + h ) Usually, as in this case, w = Putting in the mid-ordinate values we find the following. A = ( ) = units Clearly if we took more strips by say making w =.5, we would get a more accurate answer and in the limits as w becomes very small the answer will be the same as found by integrating.

4 . TRAPEZOIDAL RULE Consider that each strip has a straight line joining the top corners as shown. The height at the middle is not quite the same as the mid-ordinate and is the average of the two ordinates. If h is the average then h = (A+B)/ h = (B+ C)/ h = (C+ D)/ h = (D+E)/ The area of each strip is wh = w(a+b)/ wh = w(b+ C)/ wh = w(c+ D)/ wh = w(d+e)/ Figure The total area is A = (w/)[(a+b) + (B+C) + (C+D) + (D+E)] A = (w/)[(a+b + B+C + C+D + D+E] A = (w/)[(a + E) + (B+C+D)] Hence in our example A = (/)[(+9) +(+7+)] = (/)(+6) = This is slightly larger than the correct figure but again, if smaller strips are used, the answer will be accurate. The above rule may be written as follows.. SIMPSON'S RULE w A First Last x sum of the rest The area is divided into an even number of strips. The ordinates are h, h... The area is calculated on the assumption that the curve joining neighbouring ordinates are a quadratic that passes through the mid ordinate. It follows that if the curve is a parabola, the area will be exact. The derivation is not given here as it is quite complicated but the result is as follows. I w first last sum of the even ordinates sum of the remaining odd ordinates

5 WORKED EXAMPLE No. Find the area under the graph of the function y = sin between the limits and radians using integration and the trapezoidal rule. SOLUTION Evaluating the ordinates and mid-ordinates at intervals of /8 produces the table and graph shown. Figure 5 Integration π A sinθ dθ π - cosθ cos π cos Mid-ordinates w = / π A w(h h h h) ( ).68 Trapezoidal Rule w = / w A First Last x sum of the rest π/ A units Note one answer is slightly large and the other slightly small. 5

6 WORKED EXAMPLE No. Find the area under the curve f(x) = x + x + 8 between x = and x = using Simpson's Rule with eight strips and determine the error. SOLUTION A (x x 8)dx x A x 8x SIMPSON'S RULE () () 8() 6.67 Fig.6 w I first last sum of the even ordinates sum of the remaining odd ordinates.5 I I I The error is zero and it always is when the function is a quadratic. It follows that for a quadratic you only need two strips. SELF ASSESSMENT EXERCISE No.. Find the area under the graph of the following functions using integration, the mid-ordinate rule and the trapezoidal rule. y = x between the limits x = and x = 5 y = e x between the limits of x = and x = 5. y = sin x between the limits x = and x = 8 o. 5. Estimate the value of the definite integral I x dx by Simpson's rule using four strips. What is the error in the estimate? (65. and.5 too large) 6

7 In Engineering the area under the graph represents real things. For example the area under a force distance graph represents the work done or energy used and the area under a pressure volume graph also represents work done during the compression or expansion of a gas. WORKED EXAMPLE No. The pressure (p) and volume () during a gas expansion is related by the law p =. -.. Determine the work done when the volume is expanded from x -6 m to x -6 m. Use calculus and the trapezoidal rule to find the answer. SOLUTION Figure 7 INTEGRATION Area Work W. W. 6. x x x.69 Joules W pd but p. -. d. -. x. TRAPEZOIDAL RULE w = x -6 m w W x W W 5 x First Last x sum of the rest x.6 x x x x ( Joules -6 7

8 SELF ASSESSMENT EXERCISE No.. The electric current charging a capacitor is related to time by the following law. I = ( e -t/ ) Amps Calculate the charge Q (the area under the graph) between the limits t = and calculus and the trapezoidal rule. (Graph and ordinates are calculated for you) (Answer around Coulombs) t = 6 s. Use Figure 8. Find the area (with units) under the following function between the limits x = and x = m using integration, the mid-ordinate rule and the trapezoidal rule. (Answers around 76.7 m ) y = + x x m. Find the area under the following function between the limits t = and t = s using integration, the mid-ordinate rule and the trapezoidal rule with steps of.s. v = t + e t m/s (Answers around. m). Find the area (with units) under the following function between the limits = and =. radian using integration, the mid-ordinate rule and the trapezoidal rule with steps of. radian T = cos (Nm) (Answers around.96 Joules) 5. Find the area (with units) under the following function between the limits = and = 5 m using integration, the mid-ordinate rule and the trapezoidal rule with steps of. p = ln N/m (Answers around 8.9 Nm or Joules) 8