Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       
(b) ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves a substitution. The given integral is ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍. To simplify, let u = 𝓍³. Then, compute the derivative of u with respect to 𝓍: du/d𝓍 = 3𝓍², or equivalently, du = 3𝓍² d𝓍.
2. Step 2: Rewrite the integral in terms of u. Substitute u = 𝓍³ and du = 3𝓍² d𝓍 into the integral. The integral becomes (1/3) ∫ f(u) du, where the factor of 1/3 comes from the substitution.
3. Step 3: Adjust the limits of integration. When 𝓍 = -2, u = (-2)³ = -8. When 𝓍 = 2, u = (2)³ = 8. Therefore, the new limits of integration are from u = -8 to u = 8.
4. Step 4: Use the property of even functions. Since ƒ is an even function, ƒ(u) = ƒ(-u). This allows us to simplify the integral over symmetric limits. Specifically, ∫₋₈⁸ ƒ(u) du = 2 ∫₀⁸ ƒ(u) du.
5. Step 5: Substitute the given value of ∫₀⁸ ƒ(𝓍) d𝓍 = 9 into the equation. The integral becomes (1/3) * 2 * 9. Simplify this expression to find the final result.

Key Concepts

Even Functions

An even function is defined by the property that ƒ(−x) = ƒ(x) for all x in its domain. This symmetry about the y-axis implies that the area under the curve from -a to a is twice the area from 0 to a. Understanding this property is crucial for evaluating integrals involving even functions, as it simplifies calculations and allows for the use of symmetry.

Substitution in Integration

Substitution is a technique used in integration to simplify the process by changing the variable of integration. It involves selecting a new variable, often denoted as u, which is a function of x, and transforming the integral accordingly. This method is particularly useful when dealing with composite functions, as it can make the integral more manageable and easier to solve.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits, a and b. It is denoted as ∫ₐᵇ ƒ(x) dx and provides a numerical value that reflects the accumulation of quantities, such as area or volume. Understanding the properties of definite integrals, including their evaluation and the Fundamental Theorem of Calculus, is essential for solving problems involving area calculations and function analysis.