Problem 26 - Calculus of a Single Variable

Question

Finding the Area of a Region In Exercises \(17-30,\) sketch the region bounded by the graphs of the equations and find the area of the region. $$ f(y)=y(2-y), g(y)=-y $$

Step-by-Step Solution

Answer

The area of the region bounded by the graphs of the equations \(f(y) = y(2-y)\) and \(g(y) = -y\) from \(y = 0\) to \(y = 1.5\) is \(0.375\).

Step 1: Finding the Intersections

Set \(f(y) = g(y)\) to find the points of intersection, which are when the two functions meet. Solve the equation \(y(2-y) = -y\). We find that \(y = 0\) and \(y = 1.5\) are the solutions.

Step 2: Graphing the Functions

Plot the functions \(f(y) = y(2-y)\) and \(g(y) = -y\). The areas of interest are from \(y = 0\) and \(y = 1.5\). The functions \(f(y)\) and \(g(y)\) intersect at these points.

Step 3: Calculating the Area

To find the area between these two curves from \(y = 0\) to \(y = 1.5\), subtract the lower function from the higher one, which gives the formula: \(\int_{0}^{1.5} (f(y) - g(y))dy\). Compute the integral, resulting in \(0.375\).

Key Concepts

Integration

Understanding integration is crucial for calculating areas, volumes, and other concepts in mathematics and its applications. Integration can be thought of as the inverse process of differentiation or as a way to determine the cumulative effect of a function over an interval.

In the context of finding the area between curves, integration comes into play by adding up infinitesimally small slices or rectangles under the curve. Imagine you're slicing a loaf of bread; each slice has a thickness that approaches zero. As the slices become infinitely thin, the sum of their areas gives you the total area under the curve. When you integrate a function over a certain domain, what you're actually doing is computing the total area between the curve of the function and the x-axis within that domain.

Definite Integrals

A definite integral has both a start and an end point, also known as the limits of integration. This tool allows us to calculate the exact area under a curve between two points. When faced with a problem like the one we're discussing, the definite integral provides a numerical result representing the area of the region enclosed by the curves.

The integral is symbolized by the elongated 'S' and the limits of integration are noted as the numbers on the bottom and top of the symbol, respectively. For example, when you see \(\int_{a}^{b} f(x)dx\), it means you’re computing the area under the curve of the function f(x) from x=a to x=b. The dx at the end signifies that the integration is done with respect to x.

Sketching Regions

Before diving into calculations, it's often helpful to have a visual representation of the problem at hand. Sketching the region bounded by the curves gives us a tangible sense of what we're trying to find the area of.

Start by plotting the given functions on the coordinate plane, paying special attention to where they intersect, as these points define the limits of the region of interest. Once plotted, the enclosed region becomes apparent. This process not only aids in understanding the task but also helps prevent potential errors by confirming that the integration boundaries are correctly identified. It's like drawing a fence around the area you're interested in measuring—you wouldn't start measuring without knowing the boundaries.

Functions Intersection

The points where two curves cross each other are called intersections. These intersections are significant in problems involving the area between curves because they often dictate the boundaries for integration.

To find these points, set the equations of the two functions equal to each other and solve for the variable. The solutions to that equation, just like in our exercise, are the y-values at which the curves intersect. Once you have found these y-values, you can use them as the limits of integration to ensure you're finding the area of the correct region. This is akin to knowing the corner points of a plot of land when you're figuring out its size; without these, you wouldn't know where the land begins and ends.