Problem 77 - Calculus Volume 1

Question

In the following exercises, evaluate the integral using area formulas. \(\int_{2}^{3}(3-x) d x\)

Step-by-Step Solution

Answer

The integral evaluates to 0.5.

Step 1: Understand the Geometric Interpretation of the Integral

The integral \( \int_{2}^{3}(3-x) \, dx \) represents the signed area under the curve \( y = 3 - x \) from \( x = 2 \) to \( x = 3 \). The line \( y = 3 - x \) is a straight line with a negative slope. Between \( x = 2 \) and \( x = 3 \), this line forms a right triangle with the \( x \)-axis.

Step 2: Determine the Shape and Dimensions

The line \( y = 3 - x \) crosses the \( x \)-axis at \( x = 3 \) (since \( 3-x=0 \) when \( x=3 \)). When \( x=2 \), the value of \( y \) is \( 1 \) (so \( y=3-2 \)). This means the triangle has a base from \( x=2 \) to \( x=3 \) (length 1 unit) and a height of 1 unit at \( x=2 \).

Step 3: Apply the Formula for the Area of a Triangle

The area of a triangle is given by \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). For this triangle, the base is 1 unit, and the height is 1 unit. Therefore, the area is \( \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \).

Step 4: Determine the Sign of the Area

Since the line \( y = 3 - x \) is above the \( x \)-axis from \( x=2 \) to \( x=3 \), the area should be considered positive. Thus, the integral of \( 3-x \) from 2 to 3 gives \( \frac{1}{2} \).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in integral calculus. They help us calculate the total accumulation of something, whether it's area, volume, or any other quantity, over a specified interval on the x-axis. For instance, when you have the integral \[ \int_{a}^{b} f(x) \, dx \] it represents the accumulation of the function \( f(x) \) from \( x = a \) to \( x = b \).

• The numbers \( a \) and \( b \) are known as the limits of integration, where \( a \) is the lower limit and \( b \) is the upper limit.
• The definite integral calculates the net area, which means it may be positive, negative, or zero depending on whether the function is above or below the x-axis within the interval \( [a, b] \).

Definite integrals have practical applications, such as finding the area under curves, which you will often encounter in various mathematical and real-world problems. They are not just abstract concepts; they are essential tools for understanding how things change and accumulate over an interval.

Area Under a Curve

One of the most common uses of definite integrals is to find the area under a curve—a visual and practical application. Consider a curve defined by the function \( y = f(x) \) between two points on the x-axis, \( x = a \) and \( x = b \). The definite integral \[ \int_{a}^{b} f(x) \, dx \] provides the total area between the curve and the x-axis over that interval.

• If the function \( f(x) \) is above the x-axis, the integral gives the positive area.
• If \( f(x) \) is below the x-axis, the integral gives a negative area, indicating that it's below the reference line.

This area calculation is crucial in fields ranging from physics to economics. It can represent quantities like distance traveled, accumulated profits, or resources used, depending on the context of the function. By breaking down complex areas into simple geometrical shapes, like right triangles or rectangles, you can use familiar formulas to find exact areas, as shown in the original exercise.

Geometric Interpretation of Integrals

The geometric interpretation of integrals involves understanding how areas and shapes are formed by functions along specific intervals. When dealing with a function like \( y = f(x) \), its graph can be seen as outlining a region with the x-axis. The role of the definite integral \[ \int_{a}^{b} f(x) \, dx \] is to compute the signed area of this region.
Here are some key insights:

• The graph of \( f(x) \) determines the nature of the area. If it stays above the x-axis throughout the interval, the area is positive. If it falls below, the area is negative.
• Geometric shapes, such as triangles or rectangles, form when the function is linear or broken into segments. These shapes simplify the calculation of areas using basic geometry.
• In the original example, the linear function between \( x = 2 \) and \( x = 3 \) forms a right triangle above the x-axis, allowing for an easy area computation.

Understanding geometry's role in integrals enriches your ability to visualize and solve some integrals intuitively without solely relying on algebraic manipulation.