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)

검증된 단계별 안내

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.

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.

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.

Step 4: Substitute the value of the original integral, which is given as 32. Therefore, the integral in part (a) becomes -32.

Step 5: Conclude that the integral ∫₄⁰ 3𝓍(4 ― 𝓍) d𝓍 evaluates to -32 based on the properties of integrals and the given information.

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주요 개념

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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'.