Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.


(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍

Verified step by step guidance

1. Step 1: Recall the symmetry properties of even and odd functions. An even function satisfies ƒ(𝓍) = ƒ(−𝓍), and its integral over a symmetric interval [−a, a] is twice the integral over [0, a]. An odd function satisfies g(𝓍) = −g(−𝓍), and its integral over a symmetric interval [−a, a] is 0.
2. Step 2: Break the given integral ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍)) d𝓍 into two separate integrals: ∫₋₄⁴ 4ƒ(𝓍) d𝓍 and ∫₋₄⁴ −3g(𝓍) d𝓍. This uses the linearity property of integrals.
3. Step 3: For the first term, ∫₋₄⁴ 4ƒ(𝓍) d𝓍, note that ƒ(𝓍) is an even function. Therefore, ∫₋₄⁴ ƒ(𝓍) d𝓍 = 2∫₀⁴ ƒ(𝓍) d𝓍. Multiply this result by 4 to account for the coefficient.
4. Step 4: For the second term, ∫₋₄⁴ −3g(𝓍) d𝓍, note that g(𝓍) is an odd function. The integral of an odd function over a symmetric interval [−a, a] is 0. Therefore, this term evaluates to 0.
5. Step 5: Combine the results from Step 3 and Step 4. The final integral ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍)) d𝓍 simplifies to the result obtained from the first term, as the second term is 0.

Key Concepts

Even and Odd Functions

An even function satisfies the property f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, an odd function satisfies g(-x) = -g(x), indicating symmetry about the origin. These properties are crucial for evaluating integrals over symmetric intervals, as they allow simplifications based on the behavior of the functions.

Properties of Definite Integrals

Definite integrals have specific properties that can simplify calculations. For instance, the integral of an even function over a symmetric interval [-a, a] is twice the integral from 0 to a, while the integral of an odd function over the same interval is zero. These properties help in evaluating integrals without direct computation.

Linear Combination of Integrals

The linearity of integrals allows us to combine integrals of functions through addition and scalar multiplication. Specifically, ∫(af(x) + bg(x))dx = a∫f(x)dx + b∫g(x)dx, where a and b are constants. This property is essential for evaluating integrals involving multiple functions, as it enables the separation of terms for easier computation.