Problem 57 - Calculus for Biology and Medicine

Question

Use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{0}^{5} e^{-x} d x $$

Step-by-Step Solution

Answer

The integral \( \int_{0}^{5} e^{-x} \, dx \) represents the area under the curve of \( y = e^{-x} \) from \( x = 0 \) to \( x = 5 \).

Step 1: Identify the Function and Interval

The given integral is \( \int_{0}^{5} e^{-x} \, dx \). The function to focus on is \( f(x) = e^{-x} \), and the interval is from \( x = 0 \) to \( x = 5 \).

Step 2: Understand the Function Graphically

The function \( f(x) = e^{-x} \) is an exponential decay function, which starts at \( y = 1 \) when \( x = 0 \) and exponentially decays towards zero as \( x \) increases. This function will always stay above the x-axis for any real number.

Step 3: Sketch the Graph of the Function

Draw the graph of \( y = e^{-x} \) over the interval from \( x = 0 \) to \( x = 5 \). It starts at the point (0,1) and gets closer to the x-axis as \( x \) increases. The curve is decreasing.

Step 4: Visualize the Area Under the Curve

The area under the curve of \( y = e^{-x} \) from \( x = 0 \) to \( x = 5 \) represents the definite integral \( \int_{0}^{5} e^{-x} \, dx \). This area is bounded by the curve, the x-axis, and the vertical lines \( x = 0 \) and \( x = 5 \).

Step 5: Interpret the Integral as Area

In terms of areas, the definite integral represents the total area under the curve \( y = e^{-x} \) from \( x = 0 \) to \( x = 5 \). Since the function is always positive over this interval, this area is the actual value of the integral.

Key Concepts

Exponential Decay

The exponential decay function is an important concept when dealing with integrals. This type of function is characterized by its continuous decrease as the input value increases. In our given function \(f(x) = e^{-x}\), the decay starts at \(y = 1\) when \(x = 0\) and decreases towards zero as \(x\) becomes larger. This behavior is typical for exponential decay, where the rate of decrease is proportional to the current value, leading to a rapid decline initially that slows down over time.

• The base of the exponent in \(e^{-x}\) is the constant \(e\), which is approximately equal to 2.71828.
• This provides the function with a unique property of decreasing rapidly and never actually reaching zero.
• Observing this curve, we notice that as \(x\) increases, \(f(x)\) gets closer and closer to the x-axis.

Understanding this nature of exponential decay is crucial because it significantly impacts how we evaluate the area under the curve in the context of integrals.

Area Under the Curve

The concept of the "area under the curve" is central to understanding definite integrals. When you encounter a definite integral like \( \int_{0}^{5} e^{-x} \, dx \), you are essentially looking for the total region enclosed by the curve, the x-axis, and the vertical lines \(x = 0\) and \(x = 5\).

• This area represents the accumulation of values from the starting point \(x = 0\) to the endpoint \(x = 5\).
• Because the function \(e^{-x}\) is always positive in this interval, the "area under the curve" directly translates to the numerical value of the integral.
• The area provides a tangible representation of the integral, which can be interpreted as a sum of infinitely small increments over the interval.

This interpretation of area connects the visual representation of functions with numerical evaluations, bridging the gap between geometry and algebra.

Graphical Interpretation of Integrals

Visualizing integrals graphically can make them much more intuitive. When we draw a graph of \(y = e^{-x}\) from \(x = 0\) to \(x = 5\), the definite integral \( \int_{0}^{5} e^{-x} \, dx \) translates to the actual space that the curve covers above the x-axis.

• The graph starts at point \((0, 1)\) and slopes downward toward the x-axis while still staying above it.
• By shading the area under this curve, we can "see" the integral as an enclosed region.
• Although the function continues indefinitely, our focus is just on the segment between the two vertical lines \(x = 0\) and \(x = 5\).

This graphical interpretation allows us to understand that the integral quantifies how much space lies between the curve and the x-axis over a specified interval. It highlights the importance of visual learning when trying to comprehend abstract mathematical concepts.