Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.

∫₀⁵ (𝓍²―9) d𝓍

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Step 1: Identify the integral to be evaluated. The problem asks us to compute the definite integral ∫₀⁵ (𝓍² - 9) d𝓍 using the Fundamental Theorem of Calculus.

Step 2: Find the antiderivative of the integrand (𝓍² - 9). The antiderivative of 𝓍² is (𝓍³)/3, and the antiderivative of -9 is -9𝓍. Therefore, the antiderivative of (𝓍² - 9) is F(𝓍) = (𝓍³)/3 - 9𝓍.

Step 3: Apply the Fundamental Theorem of Calculus. This theorem states that for a continuous function f(𝓍) on [a, b], the definite integral ∫ₐᵇ f(𝓍) d𝓍 is equal to F(b) - F(a), where F(𝓍) is the antiderivative of f(𝓍). Here, we evaluate F(5) - F(0).

Step 4: Substitute the limits of integration into the antiderivative. Compute F(5) = (5³)/3 - 9(5) and F(0) = (0³)/3 - 9(0). Then subtract F(0) from F(5).

Step 5: Sketch the graph of the integrand f(𝓍) = 𝓍² - 9. This is a parabola opening upwards with its vertex at (0, -9). Shade the region between the curve and the x-axis from x = 0 to x = 5. Note that the area below the x-axis contributes negatively to the net area.

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주요 개념

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and 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 original function at the endpoints. This theorem allows us to evaluate definite integrals by finding an antiderivative of the integrand and applying the limits of integration.

Definite Integral

A definite integral represents the net area under a curve defined by a function over a specific interval [a, b]. It is calculated as the limit of Riemann sums and provides a numerical value that can be interpreted as the accumulation of quantities, such as area, over that interval. The definite integral is denoted as ∫ₐᵇ f(x) dx, where f(x) is the integrand.

Graphing the Integrand

Graphing the integrand involves plotting the function that is being integrated, which helps visualize the area under the curve. In the context of definite integrals, shading the region between the curve and the x-axis over the interval [a, b] illustrates the net area calculated by the integral. This visual representation aids in understanding the relationship between the function and its integral.

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found. ∫₀⁵ (𝓍²―9) d𝓍