Textbook Question
Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m β n.
c.
Ο
β« sin(mx) cos(nx) dx = 0, when |m + n| is even
0
3
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8. Definite Integrals
Introduction to Definite Integrals
1:58 minutesProblem 5.2.51a
Textbook Question
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(a) β«ββ° 3π(4 β π) d(π)
Verified step by step guidance1
Step 1: Recognize the integral given in part (a) is the same as the integral provided in the problem, except the limits of integration are reversed. The integral provided is β«ββ΄ 3π(4 β π) dπ = 32.
Step 2: Recall the property of definite integrals: reversing the limits of integration changes the sign of the integral. Mathematically, β«βα΅ f(π) dπ = -β«α΅β f(π) dπ.
Step 3: Apply this property to the integral in part (a). Since the limits are reversed (from 4 to 0 instead of 0 to 4), the integral becomes -β«ββ΄ 3π(4 β π) dπ.
Step 4: Substitute the value of the original integral, which is given as 32. Therefore, the integral in part (a) becomes -32.
Step 5: Conclude that the integral β«ββ° 3π(4 β π) dπ evaluates to -32 based on the properties of integrals and the given information.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as β«βα΅ f(x) dx, where 'a' and 'b' are the limits of integration. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.
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Properties of Integrals
The properties of integrals include linearity, which allows for the integration of sums and scalar multiples, and the reversal of limits, which states that β«βα΅ f(x) dx = -β«α΅β f(x) dx. These properties enable the evaluation of integrals by transforming them into simpler forms or by changing the limits of integration.
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Substitution in Integrals
Substitution is a technique used in integration to simplify the integrand by changing variables. This method involves selecting a new variable that simplifies the integral, allowing for easier computation. For example, if u = g(x), then dx can be expressed in terms of du, transforming the integral into a more manageable form.
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