Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find the area bounded by the curves \(y=e^{x}, y=1-x\), and \(x=1\).

Step-by-Step Solution

Answer

The area bounded by the curves is \(1 - e + \frac{1}{2}\).

Step 1: Find the intersection points

First, find the points where these curves intersect. Set \(e^{x}\) equal to \(1 - x\) and solve for \(x\). The solution is \(x = 0\). So the curves intersect at \(x = 0\) and also at \(x = 1\) because \(x = 1\) is one of the curves.

Step 2: Sketch the curves and region

Next, sketch the curves \(y=e^{x}, y=1-x,\) and \(x=1\). Then identify the bounded region. By noting that \(y = e^{x}\) is an exponential function and \(y = 1 - x\) is a linear function, it can be inferred that the bounded region lies between the two curves from \(x = 0\) to \(x = 1\).

Step 3: Set up the integral

Now, write an integral that represents the area of the bounded region. The area is the integral from \(a\) to \(b\) of the top function minus the bottom function. Here, the top function is \(y = 1 - x\) and the bottom function is \(y = e^{x}\), from \(x = 0\) to \(x = 1\). Thus, the integral is \(\int_{0}^{1}(1 - x - e^{x}) dx\).

Step 4: Evaluate the integral

Finally, evaluate the integral. This requires calculating the antiderivatives of \(1-x\) and \(e^{x}\), which are \(x - \frac{x^{2}}{2}\) and \(e^{x}\) respectively. The result from the fundamental theorem of calculus would be \( [x - \frac{x^{2}}{2} - e^{x}]_{0}^{1} = (1 - \frac{1}{2} - e) - (0 - 0 - 1) = 1 - e + \frac{1}{2}\).

Key Concepts

Definite Integral

Understanding the concept of a definite integral is essential when calculating the area between curves. In essence, a definite integral over a given interval represents the accumulation of the quantities measured by a function. When discussing the area between two curves, the definite integral provides a way to compute the total area under the 'top' curve and above the 'bottom' curve within the bounds of interest.

When setting up a definite integral to find the area between the two curves, you will integrate the difference of the functions representing the curves. In the given problem, the top function is the linear equation \(y = 1 - x\), while the bottom function is the exponential equation \(y = e^{x}\). The integral is constructed with limits from \(a\) to \(b\), which in the example exercise are the points of intersection or other stated bounds - in this case, \(x = 0\) to \(x = 1\). The definite integral is then expressed as \(\int_{0}^{1}(1 - x - e^{x}) dx\), which when evaluated will yield the area between these curves over the specified interval.

Intersection of Curves

Finding the intersection of curves is a critical step before computing the area between them. When two curves intersect, their y-values are equal at a particular x-value, which means their respective functions are equal at that point. For the curves \(y = e^{x}\) and \(y = 1 - x\), we set them equal to find their points of intersection: \(e^{x} = 1 - x\).

The solution to this equation gives us the x-values where the intersection occurs, and by substituting these back into the original functions, the y-values can be obtained. For the exercise, the intersection at \(x = 1\) is provided, and the intersection at \(x = 0\) was found by solving the equation. These points are essential because they act as the limits of integration (\