Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph and find the area of the region bounded below by the graph of each function and above by the \(x\) -axis from \(x=a\) to \(x=b\). $$f(x)=-1-\sqrt{x} ; a=0, b=9$$

Step-by-Step Solution

Answer

The area of the region bounded below by the graph of \(f(x)=-1-\sqrt{x}\) and above by the \(x\)-axis from \(x=0\) to \(x=9\) is 27 square units.

Step 1: Sketch the graph of the function

To sketch the function \(f(x) = -1 - \sqrt{x}\), we need to identify key features such as intercepts, the general shape of the curve, and any asymptotes. This function is a transformation of the parent function, \(y = \sqrt{x}\). The negative sign in front of \(\sqrt{x}\) reflects the graph over the \(x\)-axis, and the \(-1\) term shifts the graph downward by 1 unit. The function is only defined for all non-negative values of \(x.\) The graph will look something like this: (Here, you can refer the graph from any online graphing calculator by searching "graph of -1 - sqrt(x)")

Step 2: Find intersection points with the x-axis

To find the intersection points with the x-axis, we need to determine when the function \(f(x)\) equals 0. $$0 = -1 - \sqrt{x}$$ Solve this equation for \(x\) to find the intersection point: $$\sqrt{x} = -1$$ Since the square root of a number cannot be negative, there is no intersection point with the \(x\)-axis.

Step 3: Determine the area of the region using integration

Since there is no intersection with the x-axis, the region we're finding is completely below the x-axis, but the integral will result in a negative value. Thus, we'll find the absolute value of the integral to find the area. The definite integral to calculate the area A of the region bounded below by the graph and above by the x-axis from \(x=a\) to \(x=b\) is given by: $$A = \int_a^b \! f(x) \, \mathrm{d}x$$ $$A = \int_0^9 \! (-1-\sqrt{x}) \, \mathrm{d}x$$ Now, integrate the function with respect to \(x\) and evaluate the result for the given limits: $$A= \Big[ -x-\frac{2}{3}\sqrt[3]{x^3} \Big]_0^9$$ $$A= \Big( -9 - \frac{2}{3} (9) \Big) - \Big( 0 - \frac{2}{3}(0) \Big)$$ $$A= -27$$ Since the value of the area A is negative, take the absolute value to find the bounded area: $$A=|-27|=27$$ The area of the region bounded below by the graph of \(f(x)=-1-\sqrt{x}\) and above by the \(x\)-axis from \(x=0\) to \(x=9\) is 27 square units.

Key Concepts

Definite Integrals

In calculus, definite integrals are crucial for finding the area under a curve between two points on the x-axis. When using a definite integral, you calculate the accumulated "net" area between the curve of a function and the x-axis. This is achieved by evaluating the integral of the function over a specified interval. In our exercise, we're using the function \( f(x) = -1 - \sqrt{x} \) over the interval \([0, 9]\).

Here’s how it works:

• Identify the function and the interval. In this case, the interval is from \( x=0 \) to \( x=9 \).
• Set up the definite integral: \( \int_0^9 (-1 - \sqrt{x}) \, \mathrm{d}x \).
• Integration involves finding an antiderivative, or a function whose derivative gives the original function. For \(-1 - \sqrt{x}\), the antiderivative is \( -x - \frac{2}{3}x^{3/2} \).
• Evaluate this antiderivative at the boundaries of the interval and subtract: \( \left[ -x - \frac{2}{3}x^{3/2} \right]_0^9 \).
• The result gives the net area, which can potentially be negative, reflecting the position of the curve below the x-axis. Thus, the absolute value is commonly taken to represent geometric area, which must always be positive.

A definite integral helps in calculating the total area, which is particularly helpful in practical applications like statistics and physics.

Area under a Curve

Finding the area under a curve on a graph is a common application of integration. The curve's position relative to the x-axis determines how the calculation is performed. If the function is completely below the x-axis within the given limits, like in our function \( f(x) = -1 - \sqrt{x} \), the area derived directly from the integral would be negative.

Here's what happens step by step:

• Calculate the definite integral directly to find the "net" area between the curve and the x-axis.
• The integral from 0 to 9 of \( -1 - \sqrt{x} \) results in a negative number due to the entire curve being below the x-axis.
• To reflect the true geometric area, take the absolute value. In our problem, the definite integral gives \(-27\), and its absolute value correctly interprets the area as \(27\).
• The concept of absolute value ensures we are measuring a real-world space quantity that should always be positive.

The concept of area under a curve is crucial not only for abstract mathematics but also in disciplines like economics and physics, where interpreting these areas has real-world importance.

Graph Sketching

Sketching graphs is an essential skill in calculus to visualize the behavior of functions. The function \( f(x) = -1 - \sqrt{x} \) can be visualized by transforming the parent function \( y = \sqrt{x} \). Understanding these transformations allows us to predict how the graph will behave before performing detailed calculations.

Here's how to approach sketching:

• Start with the parent function. Here, that's \( y = \sqrt{x} \).
• Apply transformations. Negation of \( \sqrt{x} \) reflects the graph below the x-axis, producing \( y = -\sqrt{x} \).
• The addition of \(-1\) further shifts the entire curve downward by one unit, resulting in the overall function stand as \( y = -1 - \sqrt{x} \).
• Given this, the graph should start at \((0,-1)\) and decline as \(x\) increases.

For this function, the graph falls as it moves along the x-axis because of the negative square root, giving a clear image of how the curve behaves between the points of interest. Graphical analysis clarifies concepts comprehensively, offering us insight into the nature of the function's changes.