Problem 106 - Applied Calculus

Question

Find a formula for \(\int_{a} c d x .\) [Hint: No calculation necessary-just think of a graph.]

Step-by-Step Solution

Answer

The integral \( \int_{a}^{b} c \, dx = c(b - a) \).

Step 1: Understand the Exercise

The exercise requires us to find the integral of the constant function with respect to \(x\) over the interval \([a, b]\). This is represented by \( \int_{a}^{b} c \, dx \).

Step 2: Recognize the Geometric Interpretation

The integral \( \int_{a}^{b} c \, dx \) can be interpreted as the area under the horizontal line \( y = c \) from \( x = a \) to \( x = b \).

Step 3: Calculate the Area of the Rectangle

Since the function is constant, the graph from \( x = a \) to \( x = b \) forms a rectangle with height \( c \) and width \( b - a \). The area of this rectangle, which equals the integral, is given by the product of its height and width: \(c \cdot (b - a)\).

Step 4: State the Formula

Thus, the formula for the integral is \( \int_{a}^{b} c \, dx = c(b - a) \).

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. It can be thought of as a flat line on a graph where the value of the function remains the same, regardless of the input value. In our context, if you think about the function

• y = c

,
its graph is simply a horizontal line at the height c. No matter what x-value you choose, the function always outputs the same result, which is c.
Constant functions are important because they provide an excellent foundation for understanding more complex functions. In integral calculus, they help to simplify computations and visualize mathematical concepts, like the area under the curve.

Area Under the Curve

In the world of calculus, finding the "area under the curve" is a crucial concept. For a constant function, this area is particularly easy to determine, as its graph forms a rectangle.
Imagine you have a flat line represented by y = c, running from x = a to x = b. This line doesn't curve or change height, making it fairly straightforward to understand. The area under the curve, in this case, is the space captured beneath this line between the points x = a and x = b, vertically down to the x-axis.
In simpler terms, think of this as finding how much space is covered by the rectangle formed by this line. This rectangle's sides are your chosen interval

• x values: a to b,
• height: constant value c.

The task of finding the integral

• is like finding the area of this rectangle,
• which is simply the height multiplied by the width!

This concept helps in understanding how integrals work, facilitating a move to more complex problems later on.

Integral Formula

The integral formula for computing the area under a curve, particularly for a constant function, is straightforward and follows basic geometric principles. When we talk about the integral of the constant function from point a to b, written as:

• \( \int_{a}^{b} c \, dx \)

We're essentially seeking the product of the constant value c and the length of the interval

• b - a.

This formula encapsulates the beautiful simplicity of constant functions. The width of the rectangle

• spans from a to b, making it b - a,

while the height,

• which remains at a constant value of c,

just multiplies with this width to give an area (and thus an integral) of

• \( c(b - a) \).

This formula is a clear demonstration of how geometry and algebra come together in calculus to solve real-world problems effortlessly. Understanding it allows students to tackle a variety of calculus problems with confidence.