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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.2.45

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.


 โˆซโ‚€โด ฦ’(๐“) d๐“, where ฦ’(๐“) = {5      if ๐“ โ‰ค 2                                                                                                                                                                                     
                      3๐“ โ€• 1  if ๐“ > 2

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Step 1: Understand the problem. The definite integral โˆซโ‚€โด ฦ’(๐“) d๐“ involves a piecewise function ฦ’(๐“), which is defined as 5 for ๐“ โ‰ค 2 and 3๐“ - 1 for ๐“ > 2. The goal is to evaluate the integral geometrically by sketching the graph of ฦ’(๐“) and calculating the area under the curve between ๐“ = 0 and ๐“ = 4.
Step 2: Sketch the graph of ฦ’(๐“). For ๐“ โ‰ค 2, ฦ’(๐“) = 5 is a horizontal line at y = 5. For ๐“ > 2, ฦ’(๐“) = 3๐“ - 1 is a linear function with slope 3 and y-intercept -1. Plot these two segments on the graph, ensuring continuity at ๐“ = 2.
Step 3: Identify the regions under the curve. From ๐“ = 0 to ๐“ = 2, the region is a rectangle with height 5 and width 2. From ๐“ = 2 to ๐“ = 4, the region is a trapezoid formed by the linear function ฦ’(๐“) = 3๐“ - 1.
Step 4: Calculate the area of each region. For the rectangle (๐“ = 0 to ๐“ = 2), the area is given by the formula for the area of a rectangle: width ร— height. For the trapezoid (๐“ = 2 to ๐“ = 4), use the formula for the area of a trapezoid: (1/2) ร— (baseโ‚ + baseโ‚‚) ร— height, where baseโ‚ and baseโ‚‚ are the values of ฦ’(๐“) at ๐“ = 2 and ๐“ = 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 โˆซโ‚€โด ฦ’(๐“) d๐“.

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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. It is denoted as โˆซ_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 โ‰ค 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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