Textbook Question
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫₋π/₄^π/⁴ sec² x dx
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8. Definite Integrals
Introduction to Definite Integrals
2:01 minutesProblem 5.3.8
Textbook Question
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
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Understand the concept of a definite integral: A definite integral calculates the net area under a curve between two specific bounds, say from x = a to x = b. It results in a numerical value rather than a general function.
Recall the Fundamental Theorem of Calculus: The definite integral of a function f(x) from a to b is computed using an antiderivative F(x), such that \( \int_a^b f(x) \, dx = F(b) - F(a) \).
Recognize the role of the constant of integration: When finding an antiderivative, a constant of integration (C) is added because indefinite integrals represent a family of functions. However, for definite integrals, this constant cancels out during subtraction.
Perform the subtraction step: When evaluating \( F(b) - F(a) \), the constant of integration (C) appears in both F(b) and F(a). Since \( C - C = 0 \), the constant does not affect the final result of the definite integral.
Conclude why the constant can be omitted: Because the constant of integration cancels out during the subtraction process, it is unnecessary to include it when evaluating a definite integral.
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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 calculates the net area under a curve between two specified limits. It is represented as ∫[a, b] f(x) dx, where 'a' and 'b' are the bounds of integration. The result is a numerical value that reflects the accumulation of quantities, such as area, over the interval [a, b].
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Antiderivative
An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). When finding an antiderivative, we often include a constant of integration (C) because the derivative of a constant is zero, meaning multiple functions can yield the same derivative. However, this constant does not affect the evaluation of definite integrals.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫[a, b] f(x) dx = F(b) - F(a). This theorem shows that when evaluating a definite integral, the constant of integration cancels out, as it is present in both F(b) and F(a), making it unnecessary for the final result.
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