Problem 35 - Calculus and its applications

Question

Use geometry to evaluate each definite integral. \(\int_{0}^{10} \frac{1}{2} x d x\)

Step-by-Step Solution

Answer

The integral evaluates to 25.

Step 1: Identify the Function Shape

The function given is a linear function: \( f(x) = \frac{1}{2}x \), which is a straight line passing through the origin with a slope of \( \frac{1}{2} \).

Step 2: Sketch the Graph and Identify the Region

The graph of \( f(x) = \frac{1}{2}x \) is a straight line that passes through the origin (0,0) and goes through (10,5). The region under the curve from \( x = 0 \) to \( x = 10 \) forms a right triangle on the xy-plane.

Step 3: Calculate the Base and Height of the Triangle

The base of the triangle is the interval on the x-axis, which is from 0 to 10, so the base has length 10. The height of the triangle at \( x = 10 \) is the value of the function at that point, which is \( \frac{1}{2} \times 10 = 5 \).

Step 4: Calculate the Area of the Triangle

The area of a triangle is calculated as \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Substituting the base (10) and the height (5), the area under the curve is \( \frac{1}{2} \times 10 \times 5 = 25 \).

Step 5: Interpret the Integral

The definite integral \( \int_{0}^{10} \frac{1}{2} x \, dx \) represents the area under the curve of \( f(x)\) from \( x = 0 \) to \( x = 10 \), which we calculated as 25.

Key Concepts

Linear Function

A linear function is a type of mathematical function that forms a straight line when graphed. The general form of a linear function is given by the equation \( f(x) = mx + b \). Here, \( m \) is the slope, which determines the steepness of the line, and \( b \) is the y-intercept, indicating where the line crosses the y-axis. In the context of our exercise, the linear function is \( f(x) = \frac{1}{2}x \), which means:

• The slope \( m = \frac{1}{2} \) causes the line to rise half a unit on the y-axis for every one unit it moves along the x-axis.
• Since the equation does not include a \( b \), we can infer that the line crosses the origin (0,0).

Linear functions are simple yet powerful in representing relationships where one quantity increases or decreases at a constant rate relative to another. They are foundational in algebra and appear frequently in both academic studies and real-world applications.

Area Under Curve

The concept of the area under a curve is pivotal in understanding definite integrals. The definite integral of a function from \( a \) to \( b \) is essentially calculating the area between the function graph and the x-axis, bound by the vertical lines at \( x = a \) and \( x = b \). For the exercise at hand, the linear function \( f(x) = \frac{1}{2}x \) creates a right triangle when bounded by \( x = 0 \) and \( x = 10 \). To find the area under this curve:

• Identify the shape formed (here, a triangle).
• Determine the coordinates intersected by the line and the x-axis, to understand the base of the triangle.
• Calculate the height, using the y-value when \( x = 10 \).

The integral gives us the total area enclosed, which is solved geometrically in this case by the formula for the area of a triangle.

Triangle in Geometry

Triangles are fundamental shapes in geometry, often used to facilitate the calculation of areas, especially under linear functions like in this exercise. The area of a triangle is determined using the formula:\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]In the context of the linear function \( f(x) = \frac{1}{2}x \), we form a right triangle with:

• Base: the distance on the x-axis from 0 to 10, calculated simply as 10.
• Height: at \( x = 10 \), the height from the x-axis to the line is \( \frac{1}{2} \times 10 = 5 \).

This setup allows us to find the triangle's area - which in definite integrals translates directly to the integral's value.