Question

Task 4 - Integration (Maximum Mark 15) 1) The mid-ordinate rule is a rule for approximating the area under an integral. Research and use the mid- ordinate rule, with four strips, to find an approximate value of 3xet* dx -0.5 Give your answer to the nearest whole number. You must explain the method clearly. 2) Using the substitution u = x2, use integration by substitution to find the value of J-0.5 also giving your answer to the nearest whole number. You must make your method clear 3) Compare your answers and discuss which method out of 1 and 2 is the easier method, stating the benefits and limitations of each method. Notes: A video explaining this can be found here: https://www.examsolutions.net/tutorials/mid-ordinate-rule/ This part of assignment is testing that you can use integration by substitution and compare alternative methods.

Answer 1

3.5 cos 3х е** du -0.5 here b = 3.5 ne 4, a -0.5, 3.5 -(-0.5) 1 ha end points of subintervals are. a = -1/2 , 2 , 3 , { 7 / 2 = b 1 3 1 Now we evaluate the function fiu) 3n ex at the mid point of these subintervals. - flo)=0 f ) = f(1) = 3e +(-0.7707) oost, os (45+25) +12-57,3.5) f = f (2) bet - f(3) = ge? so 13s en die = 1 (0+32 +62 +9e') CS Scanned with CamScanner 73263.499094

2 3.5 I = 1 zuek die substitute u = n² dy dre 23 dre du 2re S H е“ ач 3 - 2 le du こ eu 2 undo substitution. = 22 е** I = (3.5)? 3 fo.s)? 2 2 je" (el-1) I 10 313470.00726 IcScanned with CamScanner

3 In the mid-ordinate method the integral's value 73263-499094 while with analitic method, we get jet a 313 470.00726. There is huge differ- ence in both the values given by different methods. In this particular Problem method is easier because it cau be easily handle 2 method benefit : It can solve any definite integral with a single formula. limitations. sometimes error become too large. method -2 benefit :- It gives exact solution without any error. limitations. - It cannot solve all the definite integral, also we have to apply different type of approaches forms of integrands. for different CS Scanned with CamScanner