Problem 23 - Biocalculus Calculus for the Life Sciences

Question

Sketch the region and find its area (if the area is finite). \(S=\\{(x, y) | x \leqslant 1,0 \leqslant y \leqslant e^{x}\\}\)

Step-by-Step Solution

Step 1: Understand the Region

The region described by the set \( S = \{ (x, y) | x \leqslant 1, 0 \leqslant y \leqslant e^x \} \) consists of points \((x, y)\) such that:- \( x \leq 1 \): The maximum value of \( x \) is 1.- \( 0 \leq y \leq e^x \): The minimum value of \( y \) is 0, and the maximum value of \( y \) is constrained by the exponential function \( e^x \).This region originates from the y-axis and is bounded by the line \( x = 1 \) and the curve \( y = e^x \) from x-values 0 to 1.

Step 2: Determine the Intersection Points

To find the bounds of the region, determine where the line \( x = 1 \) intersects the curve \( y = e^x \). Since \( y = e^x \) when \( x = 1 \), calculate \( e^1 = e \). At \( x = 1 \), the point of intersection is \((1, e)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric shapes and figures through algebraic equations. In this case, we are looking at a two-dimensional Cartesian coordinate system. The exercise involves understanding the region defined by certain inequalities. It involves the plane where each point is characterized by coordinates \((x, y)\).

• The inequality \(x \leq 1\) defines all points to the left of and including the line \(x = 1\). This means the region we are looking at stops at \(x = 1\).
• The inequality \(0 \leq y \leq e^x\) provides the boundaries for \(y\). The value of \(y\) starts at 0 and goes up to the value given by the exponential function \(e^x\).
• When combining these conditions, we get an area partially bounded by the vertical line \(x = 1\) and the curve defined by the function \(y = e^x\) from \(x = 0\) to \(x = 1\).

To sketch such a region, plot the curve \(y = e^x\), draw the line \(x = 1\), and shade the area beneath the curve between \(x = 0\) and \(x = 1\). This visual representation in coordinate geometry helps in further calculations such as finding the area.

Integration

Integration is a fundamental concept in calculus that involves finding the area under a curve. In this exercise, we want to find the area under the curve \(y = e^x\) and above the x-axis, bound by the lines \(x = 0\) and \(x = 1\).

• To accomplish this, we use the process of integration on the function \(e^x\) from 0 to 1.
• The integral of \(e^x\) is simply \(e^x\). Hence, to find the area, we compute \(\int_0^1 e^x \, dx\).
• After setting up the integral, evaluate it by substituting the upper limit (1) and lower limit (0) into the antiderivative, yielding \([e^x]_0^1 = e^1 - e^0 = e - 1\).

This integral gives us the exact area of the shaded region below the curve and above the x-axis, between \(x = 0\) and \(x = 1\). Integration, in this context, helps us quantify the region’s size using limits and the properties of exponential functions.

Exponential Functions

Exponential functions play a crucial role in calculus and many real-world applications, known for their unique properties and rapid growth rates. The general form of an exponential function is \(f(x) = a^x\), where \(a\) is the base and \(x\) represents the exponent. In the given exercise, we focus on the natural exponential function \(e^x\).

• It continuously increases as \(x\) increases, highlighting its rapid growth.
• The unique aspect of the function \(e^x\) is that its derivative and integral are the same, \(e^x\), making it simple to work with in calculus.
• At \(x = 0\), the function \(y = e^x\) equals 1, marking the starting point of the growth on a graph. As \(x\) approaches 1, the value \(y = e^1 = e\), designating the endpoint of our region of interest.

Understanding exponential functions and their properties is essential, as they often describe natural phenomena and financial growth. They also help in defining curves that we analyze using coordinate geometry and integration.