Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. $$ y=1+\sqrt{x}, \quad y=0, \quad x=0, \quad \text { and } \quad x=4 $$

Step-by-Step Solution

Answer

The area of the region bounded by the given equations after performing all the operations is \( A = 13.333 \) square units.

Step 1: Identify the Integrating Function

The region we intend to find the area is bounded by the x-axis (y=0), the y-axis (x=0), the vertical line given by x=4 and the curve described by y=1+sqrt(x). Thus, the area to be integrated is the function represented by the equation y=1+sqrt(x).

Step 2: Set the Limits of Integration

The limits of integration are the boundaries along the x-axis. The lower limit is where the curve y=1+sqrt(x) intersects the y-axis (x=0). The upper limit is the vertical line x=4.

Step 3: Integrate the Function

To find the area under the curve, perform the following definite integral: \[A = \int_{0}^{4} (1 + \sqrt{x})dx\] where A is the area. This results in \[ A = [x + \frac{2}{3}x^{\frac{3}{2}}]_{0}^{4}\]

Step 4: Compute the Integral

Calculate the definite integral by substituting the limits into the equation, subtracting the lower limit results from the upper limit results. Perform the calculations to obtain the value of the area A.

Key Concepts

Area between Curves

When finding the area between curves, we are essentially looking for the region that is enclosed between two boundaries. In this task, the boundaries are a curve described by the equation \( y = 1 + \sqrt{x} \) and the x-axis \( y = 0 \).

• This region is also bounded vertically by the line \( x = 4 \) and horizontally from \( x = 0 \) to \( x = 4 \).
• The area under the curve and above the x-axis is what we're interested in calculating.

To visualize, imagine drawing the curve on a graph, and shading the area directly above the x-axis starting from x = 0 to x = 4. Calculating this shaded area gives us the measure of the space between the curves.

Graphing Utility

Using a graphing utility is a helpful way to verify your mathematical results visually.

• It allows you to plot the curve \( y = 1 + \sqrt{x} \) as well as other lines that form the borders of the area you are trying to find.
• You can visually confirm the intersection points and boundaries such as where the curve meets the x-axis or where x = 4.

Graphing utilities are especially advantageous for checking the accuracy of your integration bounds and help ensure you have analyzed the right area on the coordinate plane. By visualizing, you can gain a better understanding of why your integration results in a particular area.

Integration

Integration is the key process for finding areas under curves. In this context, it involves calculating a definite integral.

• The definite integral is expressed as \( \int_{0}^{4} (1 + \sqrt{x}) dx \).
• This integral sums up an infinite number of tiny rectangles under the curve from the start point \( x = 0 \) to the endpoint \( x = 4 \).

By performing this integration, we evaluate how much the function \( y = 1 + \sqrt{x} \) sits above the x-axis over the specified interval. The resulting value represents the total area between the curve and the x-axis for these boundaries.

Mathematical Modeling

Mathematical modeling involves using equations to represent real-world phenomena or theoretical concepts, such as finding an area.

• In this scenario, the equation \( y = 1 + \sqrt{x} \) acts as a model of the curve.
• Definite integration allows us to quantify the area related to this model by providing a numerical solution.

Models can use not just integration but also differentiation, graphing, and more to solve complex problems or demonstrate behaviors. Here, the mathematical model aids in understanding how much space lies between the given curve and a specified boundary.