Explanation Given information: The integrals are ∫ 0 π sin m x sin n x d x = 0 , ∫ 0 π cos m x cos n x d x = 0 and ∫ 0 2 π sin m x cos n x d x = 0 . Formula Used: The formulas are: sin m x sin n x = 1 2 [ cos ( m x − n x ) − cos ( m x + n x ) ] cos m x cos n x = 1 2 [ cos ( m x − n x ) + cos ( m x + n x ) ] sin m x cos n x = 1 2 [ sin ( m x − n x ) + sin ( m x + n x ) ] Calculation: Consider m and n are integers with m ≠ ± n . Solve the integral ∫ 0 π sin m x sin n x d x . ∫ 0 π sin m x sin n x d x = ∫ 0 π 1 2 [ cos ( m x − n x ) − cos ( m x + n x ) ] d x = 1 2 [ sin ( m − n ) x ( m − n ) − sin ( m + n ) x ( m + n ) ] 0 π Substitute the limits. 1 2 [ sin ( m − n ) x ( m − n ) − sin ( m + n ) x ( m + n ) ] 0 π = 1 2 [ sin ( m − n ) π ( m − n ) − sin ( m + n ) π ( m + n ) − sin ( m − n ) 0 ( m − n ) + sin ( m + n ) 0 ( m + n ) ] = 1 2 [ 0 ( m − n ) − 0 ( m + n ) − 0 ( m − n ) + 0 ( m + n ) ] = 0 Solve the integral ∫ 0 π cos m x cos n x d x . ∫ 0 π cos m x cos n x d x = ∫ 0 π 1 2 [ cos ( m x − n x ) + cos ( m x + n x ) ] d x = 1 2 [ sin ( m − n ) x ( m − n ) + sin ( m + n ) x ( m + n ) ] 0 π Substitute the limits

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Videos

Question

Chapter 8.2, Problem 77E

To determine

The orthogonality relations between the equations ∫0πsinmxsinnx dx=0, ∫0πcosmxcosnx dx=0 and ∫02πsinmxcosnx dx=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.2, Problem 76E

Chapter 8.2, Problem 78E

Students have asked these similar questions

The parametric equations of the function are given asx=asin²0, y = acos). Calculate [Let: a=anumerical coefficient] dy d²y and dx dx2

A tank contains 200 gal of fresh water. A solution containing 4 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of 5 gal/min. Find the maximum amount of fertilizer in the tank and the time required to reach the maximum. Find the time required to reach the maximum amount of fertilizer in the tank. t= min (Type an integer or decimal rounded to the nearest tenth as needed.)

Thumbi Irrigation Scheme in Mzimba district is under threat of flooding. In order to mitigate against the problem, authorities have decided to construct a flood protection bund (Dyke). Figure 1 is a cross section of a 300m long proposed dyke; together with its foundation (key). Survey data for the proposed site of the dyke are presented in Table 1. Table 2 provides swelling and shrinkage factors for the fill material that has been proposed. The dyke dimensions that are given are for a compacted fill. (1) Assume you are in the design office, use both the Simpson Rule and Trapezoidal Rule to compute the total volume of earthworks required. (Assume both the dyke and the key will use the same material). (2) If you are a Contractor, how many days will it take to finish hauling the computed earthworks using 3 tippers of 12m³ each? Make appropriate assumptions. DIKE CROSS SECTION OGL KEY (FOUNDATION) 2m 1m 2m 8m Figure 1: Cross section of Dyke and its foundation 1.5m from highest OGL 0.5m…

Chapter 8 Solutions

Calculus - Standalone book

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The orthogonality relations between the equations ∫ 0 π sin m x sin n x d x = 0 , ∫ 0 π cos m x cos n x d x = 0 and ∫ 0 2 π sin m x cos n x d x = 0 .

Solution Summary: The author explains the orthogonality relations between the equations displaystyle

Videos

The orthogonality relations between the equations ∫0πsinmxsinnx dx=0, ∫0πcosmxcosnx dx=0 and ∫02πsinmxcosnx dx=0.

Want to see the full answer?

Chapter 8 Solutions