Evaluate ∫₀² 3𝓍² d𝓍 and ∫₋₂² 3𝓍² d𝓍.

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Step 1: Understand the problem. You are tasked with evaluating two definite integrals: ∫₀² 3𝓍² d𝓍 and ∫₋₂² 3𝓍² d𝓍. A definite integral calculates the area under the curve of the function within the specified limits.

Step 2: Recall the formula for the integral of a power function. The integral of 𝓍ⁿ with respect to 𝓍 is (𝓍ⁿ⁺¹)/(n+1) + C, where C is the constant of integration. For definite integrals, the constant of integration is not needed because we evaluate the function at the limits.

Step 3: Apply the formula to the function 3𝓍². The integral of 3𝓍² is (3𝓍³)/3 = 𝓍³. This simplifies the integral to ∫ₐᵇ 𝓍³ d𝓍, where 'a' and 'b' are the limits of integration.

Step 4: Evaluate the first integral ∫₀² 𝓍³ d𝓍. Substitute the upper limit (𝓍 = 2) and lower limit (𝓍 = 0) into the antiderivative 𝓍³. Compute the difference: [𝓍³]₀² = (2³) - (0³).

Step 5: Evaluate the second integral ∫₋₂² 𝓍³ d𝓍. Substitute the upper limit (𝓍 = 2) and lower limit (𝓍 = -2) into the antiderivative 𝓍³. Compute the difference: [𝓍³]₋₂² = (2³) - ((-2)³).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The limits of integration indicate the interval over which the area is calculated, and the result is a numerical value that reflects this area.

Power Rule for Integration

The Power Rule for Integration is a fundamental technique used to find the integral of polynomial functions. It states that the integral of x raised to the power n is (x^(n+1))/(n+1) + C, where n is not equal to -1. This rule simplifies the process of integrating functions like 3x², making it easier to compute definite integrals.

Symmetry in Integrals

Symmetry in integrals refers to the property that can simplify calculations, particularly when dealing with even and odd functions. An even function, f(x), satisfies f(-x) = f(x), and its integral over a symmetric interval around zero can be simplified. Conversely, an odd function satisfies f(-x) = -f(x), and its integral over a symmetric interval is zero, which can be useful in evaluating integrals like ∫₋₂² 3x² dx.

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Textbook Question

Use symmetry to explain why.

∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .