Numerical Integration.docx

NUMERICAL INTEGRATION: Trapezoidal rule Numerical integration simply means the numerical evaluation of 𝒃

integrals 𝑰 = ∫𝒂 𝒇(𝒙)𝒅𝒙 Where a and b are given and f(x) is a function given analytically by a formula or empirically by a table of values. We can represent geometrically, I as the area under the curve of f(x) between a and b as shown

Fig 1. Graphical depiction of the trapezoidal rule. Geometrically, the trapezoidal rule is equivalent to approximating the area of the trapezoid under the straight line connecting f (a) and f (b) in fig 1. Recall from geometry that the formula for computing the area of a trapezoid is the height times the average of the bases (Fig2a). In our case, the concept is the same but the trapezoid is on its side (Fig.2b). Therefore, the integral estimate can be represented as I ∼= width × average height

(a) The formula for computing the area of a trapezoid: height times the average of the bases. (b) For the trapezoidal rule, the concept is the same but the trapezoid is on its side or I ∼= (b − a) × average height where, for the trapezoidal rule, the average height is the average of the function values at the end points, or [ f (a) + f (b)]/2.

Generally; Trapezoidal 𝒙𝟎 +𝒏𝒉

𝑰=∫ 𝒙𝟎

𝒇(𝒙)𝒅𝒙 =

rule

is

given

as

𝒉 [(𝒚𝟎 + 𝒚𝒏 ) + 𝟐(𝒚𝟏 + 𝒚𝟐 + ⋯ + 𝒚𝒏−𝟏 )] 𝟐

𝒃

RULES FOR EVALUATING USING ∫𝒂 𝒇(𝒙)𝒅𝒙 TRAPEZOIDAL RULE OR SIMPSON’S RULE 𝒃 1. Compare the given definite integral, say I, with ∫𝒂 𝒇(𝒙)𝒅𝒙 to get the values of f(x) and limits a, b, 𝒃−𝒂 2. Compute h= , where n denotes the number of sub-intervals 𝒏 for the problem. 3. Find the values of y=f(x), corresponding to 𝒙𝟎 = 𝒂, 𝒙𝟏 = 𝒂 + 𝒉, 𝒙𝟐 = 𝒙𝟎 + 𝟐𝒉, … . , 𝒙𝒏 = 𝒙𝟎 + 𝒏𝒉 = 𝒃 𝒂𝒏𝒅 𝒅𝒆𝒏𝒐𝒕𝒆 𝒕𝒉𝒆𝒔𝒆 𝒗𝒂𝒍𝒖𝒆𝒔 𝒂𝒔 𝒚𝟎 , 𝒚𝟏 , 𝒚𝟐 , … . . , 𝒚𝒏. . 4. Apply the specified rule to calculate I (either Trapezoidal rule or Simpson’s rule).

By trapezoidal rule, we have 𝒙𝟎 +𝒏𝒉

𝒇(𝒙)𝒅𝒙 =

𝑰=∫ 𝒙𝟎

𝒉 [(𝒚𝟎 + 𝒚𝒏 ) + 𝟐(𝒚𝟏 + 𝒚𝟐 + ⋯ + 𝒚𝒏−𝟏 )] 𝟐 𝟏

Example: Use Trapezoidal Rule to evaluate ∫𝟎 𝒙𝟑 𝒅𝒙 considering 5 subintervals. 𝟏

Sol. Let 𝑰 = ∫𝟎 𝒙𝟑 𝒅𝒙. h=0.2, f(x) = 𝒙𝟑

Trapezoidal rule is given by 𝒉 𝑰 = [(𝒚𝟎 + 𝒚𝒏 ) + 𝟐(𝒚𝟏 + 𝒚𝟐 + ⋯ + 𝒚𝒏−𝟏 )] 𝟐 Putting the values in the above equation, we have 𝟏

𝑰 = ∫ 𝒙𝟑 𝒅𝒙 𝟎

𝟎. 𝟐 [(𝟎 + 𝟏) + 𝟐(𝟎. 𝟎𝟎𝟖 + 𝟎. 𝟎𝟔𝟒 + 𝟎. 𝟐𝟏𝟔 𝟐 + 𝟎. 𝟓𝟏𝟐)] =

x y=f(x)

0 𝒚𝟎 =𝟎

0.2 0.4 0.6 0.8 1.0 𝒚𝟏 =0.008 𝒚𝟐 =0.064 𝒚𝟑 =0.216 𝒚𝟒 =0.512 𝒚𝟓 =1

=0.26

NUMERICAL INTEGRATION: Simpson’s rule Another way to obtain a more accurate estimate of an integral is to use higher-order polynomials to connect the points. For example, if there is an extra point midway between f (a) and f (b), the three points can be connected with a parabola (Fig. 3a). If there are two points equally spaced between f (a) and f (b), the four points can be connected with a third-order polynomial (Fig.3b). The formulas that result from taking the integrals under these polynomials are called Simpson’s rules.

(a) Graphical depiction of Simpson’s 1/3 rule: It consists of taking the area under a parabola connecting three points. (b) Graphical depiction of Simpson’s 3/8 rule: It consists of taking the area under a cubic equation connecting four points. 𝟏 For Simpson’s rule, the following formula applies; 𝟑

𝒙𝟎 +𝒏𝒉

∫ 𝒙𝟎

𝒇(𝒙)𝒅𝒙 𝒉 [(𝒚𝟎 + 𝒚𝒏 ) + 𝟒(𝒚𝟏 + 𝒚𝟑 + ⋯ + 𝒚𝒏−𝟏 ) + 𝟐(𝒚𝟐 + 𝒚𝟒 𝟑 + ⋯ + 𝒚𝒏−𝟐 )] =

𝟏 𝒅𝒙

Example: Evaluate ∫𝟎

𝟏

rule taking h=0.25 𝟐 dx using Simpson’s

𝟏+𝒙

𝟑

𝟏

𝑰=∫ 𝟏

f(x) =

𝟏+𝒙𝟐

𝟎

, h=0.25

x y =f(x) 𝟏 𝒅𝒙

𝑰 = ∫𝟎

𝒅𝒙 𝒉 [(𝒚𝟎 + 𝒚𝟒 ) + 𝟒(𝒚𝟏 + 𝒚𝟐 ) + 𝟐𝒚𝟐 ] = 𝟏 + 𝒙𝟐 𝟑

𝟏+𝒙𝟐

0 𝒚𝒐 =1 =

𝟎.𝟐𝟓 𝟑

0.25 0.50 𝒚𝟏 =0.9412 𝒚𝟐 =0.8

0.75 𝒚𝟑 =0.64

1.00 𝒚𝟒 =0.5

[(𝟏 + 𝟎. 𝟓) + 𝟒(𝟎. 𝟗𝟒𝟏𝟐 + 𝟎. 𝟔𝟒) + 𝟐 × 𝟎. 𝟖]

= 0.7854

CLASSWORK 1. Given that x 4.0 4.2 4.4 4.6 4.8 5.0 5.2 lo 1.3863 1.4351 1.4816 1.5261 1.5686 1.6094 1.6487 gx 𝟓.𝟐

𝟏 𝟑

Evaluate ∫𝟒 𝒍𝒐𝒈 𝒙 𝒅𝒙 by (i) Trapezoidal Rule (ii) Simpson’s Rule ANSWERS: (i) 1.82766 (ii)1.82785 𝟒

2. Evaluate ∫𝟎 𝒆𝒙 𝒃𝒚 𝑺𝒊𝒎𝒑𝒔𝒐𝒏′ 𝒔𝑹𝒖𝒍𝒆 𝒈𝒊𝒗𝒆𝒏 𝒕𝒉𝒂𝒕 𝒆 = 𝟐. 𝟕𝟐, 𝒆𝟐 = 𝟕. 𝟑𝟗, 𝒆𝟑 = 𝟐𝟎. 𝟎𝟗, 𝒆𝟒 = 𝟓𝟒. 𝟔 ANSWER:53.87 3. A curve is drawn to pass through the points given by the following table: x 1 1.5 2 2.5 3.0 3.5 4.0 y 2 2.4 2.7 2.8 3 2.6 2.1 Estimate the area bounded by the curve, x-axis and the line x=1 and x=4. ANSWER : Area= 7.78 square units 4. The following table gives the velocity v of a particle at time t:

t(seconds) 0 2 4 6 8 10 v(m/sec) 4 6 16 34 60 94 Find the distance moved by the particle in 12 secs. ANSWER: S= 552 m. 𝝅⁄ ∫𝟎 𝟐 𝒆𝒔𝒊𝒏𝒙

5. calculate Trapezoidal rule

12 136

𝒅𝒙 correct to four decimal places by