Question

Sketch a graph of \(y=-3\) on [1,5] and use geometry to find the exact value of \(\int_{1}^{5}(-3) d x\)

Short answer

Answer: The exact value of the integral \(\int_{1}^{5}(-3)dx\) is \(-12\).

Step-by-step solution

Sketch the graph of \(y=-3\) on the interval \([1,5]\)

The function is a constant function with \(y=-3\). To sketch this, draw a horizontal line at \(y=-3\) crossing the x-axis between \(x=1\) and \(x=5\).

Determine the geometric shape formed

The graph of the constant function creates a rectangle between the x-axis and the curve on the given interval. The width of the rectangle is given by the difference between the upper and lower bound of the interval, \(5-1=4\). The height is given by the value of the constant function, \(-3\).

Calculate the area of the rectangle

When we calculate the integral, we are essentially finding the signed area under the curve. Since the geometric shape formed under the curve is a rectangle, we will calculate the area of the rectangle to find the exact value of the integral: Area of Rectangle = width × height Area of Rectangle = 4 × (-3) Area of Rectangle = \(-12\) So, the exact value of \(\int_{1}^{5}(-3)dx\) is equal to \(-12\).

Key concepts

Constant Function

A constant function is one where the output value remains the same for any input in its domain. This means the graph of a constant function is a horizontal line. In our specific example, we are looking at the function given by the equation \(y = -3\).

• For any value of \(x\) in the given interval, \(y\) will always be -3.
• The line does not cross or change direction, indicating there are no fluctuations in its value.

Understanding a constant function is important because it simplifies many mathematical operations, such as integration.

Geometry and Integrals

Geometry often aids in understanding and solving integrals, especially with constant functions. When dealing with the integration of a constant function over a defined interval, the result can be seen as the area of a simple geometric shape under the curve. In our exercise, this geometric shape is a rectangle. Here's how we interpret it:

• Width: Determined by the interval length, \(x = 5 - x = 1 = 4\).
• Height: Given by the constant value of the function, which is \(-3\).

Calculating the integral in this way involves calculating the area of the rectangle formed:- This area is calculated by multiplying the width and the height together, giving us the signed area, as the height value is negative.

Area Under the Curve

The concept of the "area under the curve" refers to finding the integral of a function on a specific interval. In essence, it evaluates the total space between the curve, the x-axis, and the interval endpoints.For a constant function, particularly a negative constant like \(-3\), this area is straightforward. We note that because the curve is below the x-axis:

• This leads to a negative area, which is reflected in the negative integral value.
• The area can be calculated directly as Width \(\times\) Height (note the negative value of the height).

Thus, the question asks us to find the exact value of the integral, which is the same as calculating the signed area, resulting in \(-12\), considering both the interval and the constant function value.