Textbook Question
63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. If m is a positive integer, then β«[0 to Ο] sin^m(x) dx = 0.
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8. Definite Integrals
Introduction to Definite Integrals
4:08 minutesProblem 5.3.51d
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.
(d) β«ββΈ 3π(4 β π) d(π)
Verified step by step guidance1
Step 1: Recognize that the integral β«ββΈ 3π(4 β π) dπ can be split into two parts: β«ββ΄ 3π(4 β π) dπ and β«ββΈ 3π(4 β π) dπ, based on the interval of integration.
Step 2: Use the given information that β«ββ΄ 3π(4 β π) dπ = 32 to evaluate the first part of the integral.
Step 3: For the second part, β«ββΈ 3π(4 β π) dπ, consider the symmetry of the function 3π(4 β π). Analyze whether the function changes sign or remains symmetric over the interval [4, 8].
Step 4: If the function is symmetric and the integral over [0, 4] is known, use properties of symmetry to determine the integral over [4, 8]. Alternatively, compute β«ββΈ 3π(4 β π) dπ directly by substitution or other methods.
Step 5: Combine the results of the two integrals, β«ββ΄ 3π(4 β π) dπ and β«ββΈ 3π(4 β π) dπ, to find the value of β«ββΈ 3π(4 β π) dπ.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral represents the signed area under a curve between two specified limits. It is denoted as β«βα΅ f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval [a, b].
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Properties of Integrals
The properties of integrals include linearity, additivity, and the ability to change limits. For instance, the integral of a sum is the sum of the integrals, and the integral from a to b can be expressed as the negative of the integral from b to a. These properties allow for simplification and manipulation of integrals to facilitate evaluation.
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Substitution Method
The substitution method is a technique used to simplify the evaluation of integrals by changing the variable of integration. By substituting a new variable, often denoted as u, the integral can be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in terms of a simpler function.
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