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 ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5 if 𝓍 ≤ 2
3𝓍 ― 1 if 𝓍 > 2

Guida verificata passo dopo passo

Step 1: Understand the problem. The definite integral ∫₀⁴ ƒ(𝓍) d𝓍 involves a piecewise function ƒ(𝓍), which is defined as 5 for 𝓍 ≤ 2 and 3𝓍 - 1 for 𝓍 > 2. The goal is to evaluate the integral geometrically by sketching the graph of ƒ(𝓍) and calculating the area under the curve between 𝓍 = 0 and 𝓍 = 4.

Step 2: Sketch the graph of ƒ(𝓍). For 𝓍 ≤ 2, ƒ(𝓍) = 5 is a horizontal line at y = 5. For 𝓍 > 2, ƒ(𝓍) = 3𝓍 - 1 is a linear function with slope 3 and y-intercept -1. Plot these two segments on the graph, ensuring continuity at 𝓍 = 2.

Step 3: Identify the regions under the curve. From 𝓍 = 0 to 𝓍 = 2, the region is a rectangle with height 5 and width 2. From 𝓍 = 2 to 𝓍 = 4, the region is a trapezoid formed by the linear function ƒ(𝓍) = 3𝓍 - 1.

Step 4: Calculate the area of each region. For the rectangle (𝓍 = 0 to 𝓍 = 2), the area is given by the formula for the area of a rectangle: width × height. For the trapezoid (𝓍 = 2 to 𝓍 = 4), use the formula for the area of a trapezoid: (1/2) × (base₁ + base₂) × height, where base₁ and base₂ are the values of ƒ(𝓍) at 𝓍 = 2 and 𝓍 = 4, respectively.

Step 5: Add the areas of the two regions to find the total area under the curve. This total area represents the value of the definite integral ∫₀⁴ ƒ(𝓍) d𝓍.

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Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the limits of integration. The value of a definite integral can be interpreted geometrically as the area between the curve and the x-axis, taking into account the regions above and below the axis.

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Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has two distinct expressions: f(x) = 5 for x ≤ 2 and f(x) = 3x - 1 for x > 2. Understanding how to evaluate piecewise functions is crucial for determining the area under the curve accurately across the specified interval.

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Geometric Interpretation of Integrals

The geometric interpretation of integrals involves visualizing the area under the curve of a function. By sketching the graph of the integrand, one can identify the regions contributing to the integral's value. This approach allows for a more intuitive understanding of the integral as the total area, rather than relying solely on algebraic methods like Riemann sums.

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