Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.


(c) ∫₄⁰ 6𝓍(4 ― 𝓍) d(𝓍)

Verified step by step guidance

1. Step 1: Recognize that the integral given in the problem, ∫₄⁰ 6𝓍(4 ― 𝓍) d𝓍, is related to the integral ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 = 32. Notice the limits of integration are reversed, and the integrand has been scaled by a factor of 2.
2. Step 2: Use the property of integrals that states reversing the limits of integration changes the sign of the integral. Specifically, ∫ₐᵇ f(𝓍) d𝓍 = -∫ᵇₐ f(𝓍) d𝓍. Apply this property to rewrite ∫₄⁰ 6𝓍(4 ― 𝓍) d𝓍 as -∫₀⁴ 6𝓍(4 ― 𝓍) d𝓍.
3. Step 3: Factor out the constant 6 from the integral using the property of integrals that allows constants to be factored out. This gives -6 ∫₀⁴ 𝓍(4 ― 𝓍) d𝓍.
4. Step 4: Recognize that ∫₀⁴ 𝓍(4 ― 𝓍) d𝓍 is equivalent to the given integral ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 divided by 3, since the integrand in the given integral is scaled by a factor of 3. Therefore, ∫₀⁴ 𝓍(4 ― 𝓍) d𝓍 = 32 / 3.
5. Step 5: Substitute ∫₀⁴ 𝓍(4 ― 𝓍) d𝓍 = 32 / 3 into the expression -6 ∫₀⁴ 𝓍(4 ― 𝓍) d𝓍 to find the value of the integral. Simplify the expression to complete the solution.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ₐᵇ 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]. Understanding this concept is crucial for evaluating integrals and applying properties related to limits.

Properties of Integrals

The properties of integrals, such as linearity, additivity, and the reversal of limits, are essential for simplifying and evaluating integrals. For instance, the linearity property states that ∫(c * f(x)) dx = c * ∫f(x) dx for a constant 'c'. Additionally, the property of reversing limits states that ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx. These properties allow for manipulation of integrals to facilitate easier computation.

Substitution in Integrals

Substitution is a technique used to simplify the evaluation of integrals by changing the variable of integration. This method involves selecting a new variable 'u' that simplifies the integrand, allowing for easier integration. For example, if u = g(x), then dx can be expressed in terms of du, transforming the integral into a more manageable form. Mastery of substitution is vital for solving complex integrals effectively.