Problem 32 - Calculus and its applications

Question

Use geometry to evaluate each definite integral. \(\int_{-1}^{4} 4 d x\)

Step-by-Step Solution

Answer

The value of the integral is 20.

Step 1: Understand the Integral

The integral \( \int_{-1}^{4} 4 \, dx \) represents the area under the horizontal line \( y = 4 \) from \( x = -1 \) to \( x = 4 \). This forms a rectangle on the coordinate plane.

Step 2: Determine the Length of the Interval

The length of the interval from \( x = -1 \) to \( x = 4 \) is calculated as the difference between these two values: \( 4 - (-1) = 5 \). This is the length of the base of the rectangle.

Step 3: Determine the Height of the Rectangle

The height of the rectangle is the constant value of the function, which is \( y = 4 \).

Step 4: Calculate the Area of the Rectangle

Use the formula for the area of a rectangle: \( \text{Area} = \text{Base} \times \text{Height} \). The base is the length of the interval (5 units), and the height is the value of the constant function (4 units): \( \text{Area} = 5 \times 4 = 20 \).

Step 5: Conclude the Integral Result

The definite integral \( \int_{-1}^{4} 4 \, dx \) evaluates to the area of the rectangle, which is 20.

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. Understanding geometry can be incredibly useful when evaluating integrals like the one given in this exercise. The definite integral \( \int_{-1}^{4} 4 \, dx \) is approached by visualizing it as the area of a geometric shape on the coordinate plane.
Here, we consider a rectangle formed between the horizontal line \( y = 4 \), the \( x \)-axis, and the vertical lines \( x = -1 \) and \( x = 4 \).
The base of this rectangle is determined by the interval on the \( x \)-axis, while the height is the value of the line at any point within this interval.
The intersection of algebra and geometry allows us to compute areas directly and swiftly.

Area under a curve

The phrase "area under a curve" often refers to the region between a function's graph and the \( x \)-axis on a given interval. For instances where the function is a constant, like \( y = 4 \), this area forms a rectangle.
This concept is pivotal because it allows us to treat integral evaluation as finding an area.
In our example, we observe a flat line, and the area beneath it up to the \( x \)-axis over the interval \(-1 \) to \( 4 \) forms the basis for calculating the integral. Recognizing this is crucial because when the function is more complex, various shapes might be involved. However, the principles remain the same.

Rectangle method

The rectangle method is a straightforward approach for evaluating definite integrals, especially when the function describes a constant. It involves computing the area of each rectangle formed by the base (aligned along the \( x \)-axis) and the height (given by the value of the function).

• The length of the base is determined by the difference of the interval limits. For this exercise, the base is 5 units, from \( x = -1 \) to \( x = 4 \).
• The constant function provides a uniform height, which is also 4 units in our problem.

This leads us to apply the area formula for a rectangle: \( \text{Area} = \text{Base} \times \text{Height} \), giving us an area of 20 which directly corresponds to the value of the integral.