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Chapter 4: Problem 31

Use geometry to evaluate each definite integral. $$\int_{0}^{2} 2 d x$$

Short Answer

Expert verified
The value of the integral is 4.

Step by step solution

01

Interpret the integral

The integral \(\int_{0}^{2} 2 \, dx\) represents the area under the curve of the function \(f(x) = 2\) from \(x = 0\) to \(x = 2\).
02

Identify the geometric shape

In this case, the graph of \(f(x) = 2\) from \(x = 0\) to \(x = 2\) forms a rectangle. The height of the rectangle is \(2\) and the base is between \(0\) and \(2\).
03

Calculate the area of the rectangle

To find the area of the rectangle, multiply its height by its base. Thus, \( \text{Area} = \text{height} \times \text{base} = 2 \times (2 - 0) = 4 \).
04

Derive the value of the integral

Since the area under the curve is 4, the value of the definite integral \(\int_{0}^{2} 2 \, dx\) is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area Under Curve
When we discuss finding the 'area under the curve', we are exploring a fundamental concept in integral calculus. The area under a curve between two points is calculated using a definite integral. This represents the accumulation of quantities, which can be visualized as the space beneath the curve and above the x-axis, within given bounds. Understanding this concept allows us to solve problems in physics, engineering, and economics where we need to evaluate gathered quantities over a range.
Geometric Interpretation
To find the value of a definite integral, sometimes we can use a geometric interpretation. This involves visualizing the function graphically. For example, the function given in the exercise is a constant function, represented by a horizontal line at y = 2. The area under this constant function between x = 0 and x = 2 forms a geometric shape鈥攁 rectangle in this case. Recognizing the shape helps simplify the calculation process since we can quickly apply basic geometric area formulas.
Integral Calculation
Integral calculation can be introduced in several ways. In the provided exercise, the integral \(\backslash int_{0}^{2} 2 \textrm{d} x\) is calculated by interpreting the integral. It essentially tells us to accumulate the value of the function 2 over the interval from x = 0 to x = 2. Calculating this through elementary geometry simplifies the process. Here's how it's done: identify the shape under the curve (rectangle), calculate its area using the formula for the area of a rectangle, and then summarize the result as the value of the integral.
Rectangle Area
In geometry, the area of a rectangle is given by multiplying its base by its height. This principle is directly applied in evaluating the definite integral in the problem. The function f(x) = 2 forms a rectangle from x = 0 to x = 2. The base of the rectangle is the range of x-values (2 - 0 = 2), and the height is the constant value of the function, which is 2. Hence, the area of the rectangle, and therefore the value of the integral, is determined by calculating: Area = height \(\times\) base = 2 \(\times\) 2 = 4.

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