Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁸ 8𝓍¹/³ d𝓍

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Step 1: Recognize that the integral ∫₁⁸ 8𝓍¹/³ d𝓍 is a definite integral, and we will use the Fundamental Theorem of Calculus to evaluate it. The Fundamental Theorem states that if F'(𝓍) = f(𝓍), then ∫ₐᵇ f(𝓍) d𝓍 = F(b) - F(a).

Step 2: Identify the function to integrate, which is f(𝓍) = 8𝓍¹/³. To find the antiderivative, recall the power rule for integration: ∫𝓍ⁿ d𝓍 = (𝓍ⁿ⁺¹)/(n+1) + C, where n ≠ -1.

Step 3: Apply the power rule to the term 𝓍¹/³. The antiderivative of 𝓍¹/³ is (𝓍⁴/³)/(4/3) = (3/4)𝓍⁴/³. Multiply this by the constant 8 to get the antiderivative of the entire function: F(𝓍) = 8 * (3/4)𝓍⁴/³ = 6𝓍⁴/³.

Step 4: Use the Fundamental Theorem of Calculus to evaluate the definite integral. Substitute the upper limit (𝓍 = 8) and the lower limit (𝓍 = 1) into the antiderivative F(𝓍). This gives F(8) - F(1), where F(𝓍) = 6𝓍⁴/³.

Step 5: Write the expression for the result: F(8) - F(1) = 6(8⁴/³) - 6(1⁴/³). Simplify each term separately to find the final value of the definite integral.

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, stating that if a function is continuous on an interval [a, b], then the integral of its derivative over that interval equals the difference in the values of the function at the endpoints. This theorem allows us to evaluate definite integrals by finding an antiderivative of the integrand.