Approximating Areas
Approximating areas involves using various methods, such as the trapezoidal rule or Simpson's rule, to estimate the area under a curve or between two curves. This is particularly useful when the exact area cannot be easily calculated using traditional methods. By dividing the area into smaller, more manageable shapes and summing their areas, a close approximation can be obtained.
6 Key excerpts on "Approximating Areas"
eBook - ePub- Sandra Lee(Author)
- 2020(Publication Date)
- Wiley-Blackwell(Publisher)
...a line is drawn along the irregular or curved boundary in such a position that, so far as can be judged, the area of paving excluded by this line is equal to the area included beyond the boundary. In Figure A.1 the area of paving to be measured is enclosed by firm lines, the method of forming two triangles (the sum of which equals the whole area) being shown by broken lines. For a more accurate calculation of the irregular area, particularly if evenly spaced offsets are available dividing the area into an even number of strips, Simpson's rule may be applied. The intermediate offsets should be numbered as it is necessary to distinguish the odd numbers from the even. The formula is given on page 300. Fig. A.1 Fig. A.2 Wher e two sides of a four‐sided figure are parallel to form a trapezoid, it is not necessary to divide the figure into triangles, as the area equals the length of the perpendicular between the parallel sides multiplied by the mean length between the diverging sides. In Figure A.2 the area is EF × GH, GH being drawn halfway between AB and CD and being equal to i.e. the average of AB and CD. Another irregular figure that often puzzles the beginner is the additional area to be measured where two roads meet with the corners rounded off to a quadrant or bellmouth. This is most easily calculated as a square on the radius with a quarter circle deducted. For example, consider Figure A.3. Fig. A.3 Measurement of arches In the case of segmental arches, the deduction above the springing line and the girth of the arch are not usually calculated precisely, as they can be estimated sufficiently accurately, the former from a triangle with compensating lines or by taking an average height, and the latter by stepping the girth round with dividers. In the case of expensive work, one should be as accurate as possible, the measurements preferably being worked out by calculation...
eBook - ePub- KJ Rawson, E. C. Tupper(Authors)
- 2001(Publication Date)
- Butterworth-Heinemann(Publisher)
...2.25 The squared paper approach TRAPEZOIDAL RULE A trapezoid is a plane four-sided figure having two sides parallel. If the lengths of these sides are y 1 and y 2 and they are h apart, the area of the trapezoid is given by A = 1 2 h (y 1 + y 2) Fig. 2.26 A trapezoid A curvilinear figure can be divided into a number of approximate trapezoids by covering it with n equally spaced ordinates, h apart, the breadths at the ordinates in order being y 1, y 2, y 3,..., y n. Commencing with the left-hand trapezoid, the areas of each trapezoid are given by 1 2 h (y 1 + y 2) 1 2 h (y 2 + y 3) 1 2 h (y 3 + y 4) … Fig. 2.27 Curvilinear figure represented by trapezoids By addition, the total area A of the figure is given by A = 1 2 h (y 1 + 2 y 2 + 2 y 3 + ⋯ + y n) = h (1 2 y 1 + y 2 + y 3 + ⋯ + 1 2 y n) This is termed the Trapezoidal Rule. Clearly, the more numerous the ordinates, the more accurate will be the answer. Thus, to evaluate the expression A = ∫ y d x the shape is divided into evenly spaeed seetions h apart, the ordinates measured and substituted in the rule given above. If the ordinates represent eross-seetional areas of a solid, then the integration gives the volume of that solid, ∇ = ∫ A d x. Expressions can be dedueed for moments, but these are not as convenient to use as those that follow. SIMPSON’S RULES Generally known as Simpson’s rules, these rules for approximate integration were, in fact, dedueed by other mathematicians many years previously. They are a special case of the Newton–Cotes’ rules. Let us deduee a rule for integrating a curve y over the extent of x. It will be convenient to choose the origin to be in the middle of the base 2 h long, having ordinates y 1, y 2 and y 3. The choice of origin in no way affeets the results as the student should verify for himself. Assume that the curve can be represented by an equation of the third order, y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 The area under the curve is given...
eBook - ePub- Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...CHAPTER 11 Areas (1) Learning outcomes: (a) Calculate the areas of triangles, quadrilaterals and circles (b) Identify and use the correct units (c) Solve practical problems involving area calculation 11.11 Introduction Area is defined as the amount of space taken up by a two-dimensional figure. The geometrical properties of triangles, quadrilaterals and circles have been explained in Chapter 10. A summary of the formulae used in calculating the areas and other properties of these geometrical shapes is given in Table 11.1. The units of area used in metric systems are: mm 2, cm 2, m 2 and km 2. Table 11.1 Shape Area and other properties Area = l × b Perimeter = 2 l + 2 b = 2(l + b) Area = l × l = l 2 Perimeter = 4 l Area = l × h Area = π r 2 Circumference = 2π r 11.2 Area of triangles There are many techniques and formulae that can be used to calculate the area of triangles. In this section we consider the triangles with known measurements of the base and the perpendicular height, or where the height can be calculated easily. Example 11.1 Find the area of the triangles shown in Figure 11.1. Figure 11.1 Solution: (a) Base BC = 8 cm We need to calculate height AD, which has not been given. As sides AB and AC are equal, BD must be equal to DC. Therefore, BD = DC = 4 cm. Now we can use Pythagoras’ Theorem to calculate height AD : Therefore (b) 11.3 Area of quadrilaterals A plane figure bounded by four straight lines is called a quadrilateral. The calculation of area of some of the quadrilaterals is explained in this section. Example 11.2 Find the area of the shapes shown in Figure 11.2. Figure 11.2 Solution: (a) Area of a rectangle = length × width Length = 15 cm, and width = 6 cm Area of rectangle ABCD = 15 × 6 = 90 cm 2 (b) In a square, the length is equal to the width...
eBook - ePub- Raymond Paul, Walter Whyte(Authors)
- 2012(Publication Date)
- Routledge(Publisher)
...The ruled sheet is placed over the area to be measured and estimated give and take lines drawn in as in Figure 14.5. Knowing the distance between the ordinates, to scale, the distance between the vertical give and take lines may be scaled off for each block and the area of each block calculated as a simple rectangle. The sum of the areas of the rectangles will give an estimate of the area of the whole figure. 14.3.3.2 Mean ordinate rule In this method, a line is drawn through the centre of the area to be measured, as in Figure 14.6, the line length being d. The line is divided into equal intervals, of length / and ordinates drawn at right angles to the line, the length of the ordinates being scaled as o 1, o 2, o 3,... o n. Figure 14.6 The mean ordinate rule states that the area is equal to the mean ordinate length by the total length of the line. area = (d/n)(o x + o 2 + o 3 +... + o„) (14.9) where d is the total line length, n is the number of ordinates and the ordinates are O 1, O 2,. O 3,. . .. The method is rapid, but not very accurate. 14.3.3.3 Trapezoidal rule In this method, the shape formed between each pair of ordinates is considered to be a trapezium, then summing the area of each trapezium gives area = (l/2)(o 1 + 2o 2 + 2o 3 + 2o 4 +... 2o (n-1) + o n) (14.10) If the boundary is curvilinear then the area is a good approximation and the accuracy may be increased by increasing the number of ordinates. 14.3.3.4 Simpson's rule This is very similar to the trapezoidal method, but assumes that the irregular boundary consists of a series of parabolic arcs between the ordinates rather than straight lines. In this case, however, the area must be divided into an even number of strips by an odd number of ordinates. If there are an odd number of strips, then the last strip area must be calculated separately and added to the area calculated for the even number of strips. area = (l /3)(o 1 + 4o 2 + 2o 3 + 4o 4 +.....
- Bernard Liengme(Author)
- 2008(Publication Date)
- Academic Press(Publisher)
...Chapter 13 Numerical Integration Publisher Summary This chapter discusses numerical integration, which is used to evaluate a definite integral when there is no closed-form expression for the integral or when the explicit function is not known and the data is available in tabular form only. Numerical integration (or quadrature) consists of methods to find the approximate area under the graph of the function f(x) between two x-values. The simplest of these methods uses the trapezoid rule, which is illustrated in this chapter. Numerical Integration Numerical integration is used to evaluate a definite integral when there is no closed-form expression for the integral or when the explicit function is not known and the data is available in tabular form only. Numerical integration (or quadrature) consists of methods to find the approximate area under the graph of the function f(x) between two x -values. The simplest of these methods uses the trapezoid rule. If we divide the area under the curve into a sufficiently large number of parts, as shown in Figure 13.1, then the area under the curve (the approximate integral) is given by: Figure 13.1 We approximate the representative strip to a trapezoid. For a clearer drawing, only five strips are used. Obviously, more, smaller, strips are needed for a good approximation. Let there be n strips and hence n+1 data points. The area of a typical strip is given by: (13.2) Combining the two equations we see that: (13.3) Giving: (13.4) or (13.5) Equation 13.5 is called the trapezoid rule. We use the trapezoid approximation in Exercise 1 to evaluate an integral. A better approximation to the integral is obtained by taking two adjacent strips and joining the three points on the curve with a parabola. This gives Equation 13.6, called the Simpson ⅓ rule for approximating the area under a curve. (13.6) This rule requires that there be an even number of equally spaced strips...
eBook - ePubProject Surveying
Completely revised 2nd edition - General adjustment and optimization techniques with applications to engineering surveying
- Peter Richardus(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...as 2 A = [ (X n +X n+1) (Y n+1 − Y n) ]. (2.12) It can easily be corroborated that 2 A = [ (X n+1 − X n) (Y n − Y n+1) ], (2.13) counting anticlockwise. These formulae (2.12) and (2.13) are known as the trapezoidal formulae, since each product represents the double area of a trapezoid, the area of the polygon being the sum of these. It may seem more convenient to apply a translation of the origin of the coordinate system to the point A of the triangle ABC in Fig. 2.1. This is only ostensibly so since the machine takes care of the coordinate differences automatically. The numerous graphical methods of calculation of areas will not be treated. Attention is drawn only to the development of electronic planimeters and digitizers, which will show their economy especially in projects where many areas are to be determined. 2.2 Problems 1. Calculate the area enclosed by the polygon given by the coordinates of following points: 2. Calculate the area of the block of land, shown in the Fig. 2.4, obtained by a chain survey. The chainage along the traverse line AB and the lengths of the offsets at right angles to this line, to the boundary points are as given in the figure. Fig. 2.4...
