Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

∫₀⁴ (8―2𝓍) d𝓍

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Step 1: Recognize that the integrand, 8 - 2𝓍, represents a linear function. The graph of this function is a straight line with a y-intercept of 8 and a slope of -2. Sketch the graph of this line over the interval [0, 4].

Step 2: Identify the region enclosed by the graph of the function, the x-axis, and the vertical lines x = 0 and x = 4. This region forms a trapezoid.

Step 3: Break the trapezoid into simpler geometric shapes if necessary. In this case, the trapezoid can be analyzed directly using the formula for the area of a trapezoid: Area = (1/2) × (Base₁ + Base₂) × Height.

Step 4: Calculate the dimensions of the trapezoid. Base₁ is the value of the function at x = 0, which is 8. Base₂ is the value of the function at x = 4, which is 0. The height of the trapezoid is the distance between x = 0 and x = 4, which is 4.

Step 5: Substitute the dimensions into the formula for the area of a trapezoid to find the value of the definite integral. Interpret the result as the total area under the curve from x = 0 to x = 4.

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Key Concepts

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

Definite Integrals

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 Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the total accumulation of the quantity represented by the function between the limits of integration.

Geometric Interpretation

The geometric interpretation of definite integrals involves visualizing the area between the curve of the integrand and the x-axis over the specified interval. This area can be positive or negative depending on whether the curve lies above or below the x-axis. Understanding this concept allows for evaluating integrals using geometric shapes, such as rectangles, triangles, or trapezoids, rather than relying solely on algebraic methods.

Sketching Graphs

Sketching the graph of the integrand is crucial for visualizing the function's behavior over the interval of integration. It helps identify key features such as intercepts, maxima, minima, and the overall shape of the curve. A well-drawn graph aids in understanding the area to be calculated and can simplify the evaluation of the definite integral by allowing for geometric reasoning.

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