Lesson 10 jdworakowski（课10 jdworakowski）.docVIP

Lesson 10 Introduction to Calculus Area under the curve (estimation) There are several practical situations where it is necessary to estimate the area of irregular figures. Example include estimation of areas of plots of land by surveyors, area of indicator diagrams of steam engines by engines and area of water planes and transverse sections of a ship by naval architects. There are many methods whereby the area of an irregular plane surface. A surface may be found from. 1. Trapezoidal rule 2. Midordinate rule 3. Simpsons rule Trapezoidal rule To find the area under curve and xaxes as shown in Fig. 10.1, the base DC is divided into a number of equal intervals of with d. This can be any number the greater the number, the more accurate the result. Fig.10. An ordinates y1, y2, y3 . . . .y10 are accurately measured. The approximation used in this rule is to assume that each strip is equal to the area of a trapezium AAD D The area of a trapezium is equal to sum of the parallel sides, times one of second (1/2) perpendicular distance between the parallel sides. Hence for the first strip, shown in Fig.10.1 the approximate area is: A1= ?( y1 + y2) *d For the second strip, the approximate area is A2= ?( y2 + y3) *d , and so on. Hence the approximate area of ABCD = A1 + A2 + A3 +. . . A10 1/2(y1 + y2) * d + 1/2(y2 + y3) * d + 1/ 2(y3 + y4)*d + . .1/2 (y9 + y10) *d = d* [1/2y1 +1/2 y2 + 1/2 y2 + 1/2 y3 + 1/2 y3 +1/2 y4 + . . .1/2 y9 + + 1/2y10 ] Inside of the bracket only 1/2y1 and 1/2y10 doesn’t has a pair and then Area = d*[1/2 y1 + y2 + y3 + y4 + . . . +1/2 y10] = = d*[ + y2 + y3 + y4 + . . . y9] Generally, the trapezoidal rule states that the area of an irregular figure is given by: Area = (width of interval)*[1/2((first + last ordinate) + sum of remaining ordinates] Area = d*[(y2 + y3 + y4 + . . . .yn-1)] (10.1.) Example 1 An indicator diagram of a steam engine is 9.00 cm long. Seven evenly spaced ordinates, including the end ordina