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.
(a) ∫₄⁰ 3𝓍(4 ― 𝓍) d(𝓍)

Verified step by step guidance

1. Step 1: Recognize the integral given in part (a) is the same as the integral provided in the problem, except the limits of integration are reversed. The integral provided is ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 = 32.
2. Step 2: Recall the property of definite integrals: reversing the limits of integration changes the sign of the integral. Mathematically, ∫ₐᵇ f(𝓍) d𝓍 = -∫ᵇₐ f(𝓍) d𝓍.
3. Step 3: Apply this property to the integral in part (a). Since the limits are reversed (from 4 to 0 instead of 0 to 4), the integral becomes -∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍.
4. Step 4: Substitute the value of the original integral, which is given as 32. Therefore, the integral in part (a) becomes -32.
5. Step 5: Conclude that the integral ∫₄⁰ 3𝓍(4 ― 𝓍) d𝓍 evaluates to -32 based on the properties of integrals and the given information.

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫ₐᵇ f(x) dx, where 'a' and 'b' are the limits of integration. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.

Properties of Integrals

The properties of integrals include linearity, which allows for the integration of sums and scalar multiples, and the reversal of limits, which states that ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx. These properties enable the evaluation of integrals by transforming them into simpler forms or by changing the limits of integration.

Substitution in Integrals

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