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


∫₁⁸ 8𝓍¹/³ d𝓍

Verified step by step guidance

1. 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).
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the limits of integration, which specify the interval, and provides a numerical value that reflects the accumulation of quantities, such as area, volume, or total change, over that interval.

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In the context of the Fundamental Theorem of Calculus, finding the antiderivative is essential for evaluating definite integrals, as it allows us to compute the integral by substituting the limits of integration into the antiderivative and calculating the difference.