Problem 53 - Calculus: One and Several Variables

Question

Find the centroid of the region under the graph. $$f(x)=e^{x}, \quad x \in[0,1]$$

Step-by-Step Solution

Answer

The centroid of the region under the graph of the function \(f(x) = e^x\) for the interval \(x \in [0, 1]\) is: \[ (\bar{x}, \bar{y}) = \left(\frac{1}{e - 1}, e - 1\right) \]

Step 1: Find the Area under the Curve

To find the area under the curve, we need to integrate the given function \(f(x) = e^x\) over the interval [0, 1]. Let's calculate the definite integral: \[ A = \int_0^1 e^x \, dx \] To find the integral of \(e^x\), we know that: \[ \int e^x \, dx = e^x + C \] So, applying the limits of integration: \[ A = \left[e^x\right]_0^1 = e^1 - e^0 = e - 1 \]

Step 2: Calculate the x-coordinate of the Centroid

To find the x-coordinate of the centroid, we'll calculate the average value of x: \[ \bar{x} = \frac{1}{A} \int_0^1 x \cdot e^x \, dx \] To find the integral of \(x \cdot e^x\), we use integration by parts: Let u = x, dv = e^x dx Then du = dx, v = e^x \[ \int x \cdot e^x \, dx = x \cdot e^x - \int e^x \, dx = x \cdot e^x - e^x + C \] Now, applying the limits of integration: \[ \bar{x} = \frac{1}{e - 1} \left[\left(x \cdot e^x - e^x\right) |_0^1 \right] = \frac{1}{e - 1} \left[\left(1 \cdot e^1 - e^1\right) - \left(0 \cdot e^0 - e^0\right) \right] = \frac{1}{e - 1} (1) \]

Step 3: Calculate the y-coordinate of the Centroid

To find the y-coordinate of the centroid, we'll calculate the average value of the function: \[ \bar{y} = \frac{1}{1} \int_0^1 f(x) \, dx = \frac{1}{1} (e - 1) = e -1 \] So, the y-coordinate of the centroid is \(\bar{y} = e-1\).

Step 4: Form the Centroid

The centroid of the region under the graph of the function \(f(x) = e^x\) for the given interval is the point \((\bar{x}, \bar{y})\). Therefore, the centroid of the region under the graph is: \[ (\bar{x}, \bar{y}) = \left(\frac{1}{e - 1}, e - 1\right) \]

Key Concepts

Definite Integral

A definite integral is a powerful tool in calculus used to calculate the area under a curve within a specific interval. When you see the notation \( \int_a^b f(x) \, dx \), it means you're computing the accumulated area from \(x = a\) to \(x = b\) for the function \(f(x)\). This process helps determine various properties like the area, volume, or even the average value of a function over an interval.

In our exercise, the definite integral \( \int_0^1 e^x \, dx \) represents the area under the curve of the exponential function \(e^x\) from 0 to 1. Calculating this, we find that the area is \(e - 1\). This integral gives us a "total" view of how the function behaves between the limits, capturing the essence of the region's size or extent under the curve.

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation. It's used when you're faced with the integral of a product of two functions. The formula for integration by parts is given by:

• \( \int u \, dv = uv - \int v \, du \)

This formula unfolds an integral into parts that might be easier to evaluate separately.

In solving our problem, to find the x-coordinate of the centroid, we need to integrate \(x \cdot e^x\). Here, by choosing \(u = x\) and \(dv = e^x \, dx\), we get \(du = dx\) and \(v = e^x\). Applying the integration by parts formula, \( \int x \cdot e^x \, dx = x \cdot e^x - \int e^x \, dx \), and computing it within the interval, helped us isolate and solve for \(\bar{x}\).

Thus, integration by parts reduced our complex problem into simpler, computable expressions.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent, most commonly base \(e\), where \(e\) is Euler's number approximately equal to 2.718. This type of function, \(f(x) = e^x\), is notable for its continuous growth rate, manifesting in various aspects of mathematics, physics, and growth models in nature and finance.

The exponential \(e^x\) is unique because its derivative, \(\frac{d}{dx}e^x = e^x\), equals itself. This property makes handling calculus-related operations straightforward, such as integration and differentiation.

In our exercise, \(e^x\) smoothly describes the curve, and its properties simplify integration, ultimately assisting us in determining the centroid under this specific curve.

Area Under a Curve

The concept of the area under a curve is central to integration. It represents the "total" accumulated value enclosed by the curve \(y = f(x)\), the x-axis, and the boundaries of integration (usually defined limits). This idea is foundational in various fields such as physics for displacement and economics for total profit or cost.

In our example, the area \(A\) under the curve \(e^x\) from \(x = 0\) to \(x = 1\) was computed with a definite integral, resulting in \(e - 1\). Despite the exponential function potentially rising sharply, the definite integral accommodated this change, presenting a concise summary of the region's area. Understanding this concept is essential for calculating centroids since it serves as the stepping stone to finding coordinates of these centers of mass or balance for defined regions.