![Rectangular Rule
Rectangular Rule
⇒ Rectangular rule is the simplest rule.
⇒ We divide the integration interval uniformly in n segments.
⇒ Let h is the resolution, where
h =
b − a
n
(2)
⇒ The definite integral of f (x) between the interval a ≤ x ≤ b is
J =
Z b
a
f (x)dx ≈ h[f (x∗
1 ) + f (x∗
2 ) + ..... + f (x∗
n )] (3)
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 4 / 9](https://image.slidesharecdn.com/nmup-6-210920091146/85/Numerical-Integration-Trapezoidal-Rule-4-320.jpg)
![Trapezoidal Rule
Key Points
⇒ The trapezoidal rule is generally more accurate.
⇒ Integral is obtained by the same subdivision as before.
⇒ Approximate f by a broken line of segments (chords) with endpoints
[a, f (a)], [x1, f (x1)], ...., [b, f (b)] on the curve of f (x).
⇒ Then the area under the curve of f between a and b is approximated
by n trapezoids of areas
1
2[f (a) + f (x1)]h, 1
2[f (x1) + f (x2)]h, ..... ,1
2[f (xn−1) + f (b)]h
Geometrical Interpretation of Trapezoidal Rule:
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 6 / 9](https://image.slidesharecdn.com/nmup-6-210920091146/85/Numerical-Integration-Trapezoidal-Rule-6-320.jpg)
Numerical Integration: Trapezoidal Rule
- 1. Numerical Integration: Trapezoidal Rule Dr. Varun Kumar Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 1 / 9
- 2. Outlines 1 Introduction to Numerical Integration 2 Trapezoidal Rule 3 Example Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 2 / 9
- 3. Introduction to Numerical Integration Important points ⇒ Numerical integration means the numerical evaluation of integrals J = Z b a f (x)dx = F(b) − F(a), where F0 (x) = f (x) (1) a and b are given. f (x) is a function given analytically. J is the area under curve of f (x) between a and b. Geometrical Interpretation of a Definite Integral Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 3 / 9
- 4. Rectangular Rule Rectangular Rule ⇒ Rectangular rule is the simplest rule. ⇒ We divide the integration interval uniformly in n segments. ⇒ Let h is the resolution, where h = b − a n (2) ⇒ The definite integral of f (x) between the interval a ≤ x ≤ b is J = Z b a f (x)dx ≈ h[f (x∗ 1 ) + f (x∗ 2 ) + ..... + f (x∗ n )] (3) Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 4 / 9
- 5. Geometrical Interpretation of Rectangular Rule Figure: Rectangular rule Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 5 / 9
- 6. Trapezoidal Rule Key Points ⇒ The trapezoidal rule is generally more accurate. ⇒ Integral is obtained by the same subdivision as before. ⇒ Approximate f by a broken line of segments (chords) with endpoints [a, f (a)], [x1, f (x1)], ...., [b, f (b)] on the curve of f (x). ⇒ Then the area under the curve of f between a and b is approximated by n trapezoids of areas 1 2[f (a) + f (x1)]h, 1 2[f (x1) + f (x2)]h, ..... ,1 2[f (xn−1) + f (b)]h Geometrical Interpretation of Trapezoidal Rule: Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 6 / 9
- 7. Continued– ⇒ By taking their sum we obtain the trapezoidal rule J = Z b a f (x)dx ≈ h 2 h f (a)+2 f (x1)+f (x2)+...+f (xn−1) +f (b) i (4) where, h = b−a n Example Q Evaluate the following definite integral J with n = 10, where J = Z 1 0 e−x2 dx ⇒ Solve the above integral using rectangular rule. ⇒ Solve the above integral using trapezoidal rule. Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 7 / 9
- 8. Solution Using Trapezoidal Rule Solution: According to question, a = 0, b = 1, n = 10 and h = 1−0 10 = 0.1 J ≈ h 2 h f (a)+2 Pn−1 i=1 f (xi )+f (b) i = 0.1 2 h 1+2×6.778167+0.367879 i = 0.7617562 Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 8 / 9
- 9. Integration accuracy Solve the standard integral and check its accuracy using rectangular and trapezoidal rule 1 R 1 0 x2dx 2 R 3 1 1 1+x dx 3 R π/2 0 cos x sin5 x dx 4 R 2 1 x2e2x dx 5 R 11 4 x3 1+x5 dx 6 R 1 0 x cos xdx Note: Students can also observe the effect of n on definite integral accuracy. Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-1 9 / 9