Question

Find the area of the region in the first quadrant between the curve \(y=e^{-6 x}\) and the \(x\) -axis.

Short answer

The area is \(\frac{1}{6}\).

Step-by-step solution

Identify the Area to Integrate

The goal is to find the area under the curve \(y = e^{-6x}\) that lies in the first quadrant. This means we are interested in the area from \(x = 0\) to \(x\) where the curve intersects the \(x\)-axis, but since \(y = e^{-6x}\) will never intersect the \(x\)-axis in the first quadrant (as \(e^{-6x}\) asymptotically approaches zero), we'll consider from \(x = 0\) to infinity.

Set Up the Integral

The area under a curve from \(x = a\) to \(x = b\) is found by evaluating the definite integral of the function over that interval. For the curve \(y = e^{-6x}\), we set up the integral as \(\int_{0}^{\infty} e^{-6x} \, dx\).

Compute the Indefinite Integral

First, we find the indefinite integral of \(e^{-6x}\). The antiderivative of \(e^{ux}\) is \(\frac{1}{u} e^{ux}\). Thus, the indefinite integral of \(e^{-6x}\) is \(-\frac{1}{6} e^{-6x}\).

Evaluate the Definite Integral

We now evaluate \(-\frac{1}{6} e^{-6x}\) from \(x = 0\) to \(x = \infty\):\[\left. -\frac{1}{6} e^{-6x} \right|_{0}^{\infty} = \lim_{{b \to \infty}}\left[-\frac{1}{6} e^{-6b} - \left(-\frac{1}{6} e^{0}\right)\right]\]Evaluating this limit, \(\lim_{{b \to \infty}}[-\frac{1}{6} e^{-6b}] = 0\) (because \(e^{-6b}\) approaches 0 as \(b\) approaches infinity). Thus, the integral simplifies to:\[0 - \left(-\frac{1}{6}\right) = \frac{1}{6}\]

Confirm the Solution

The computed integral value \(\frac{1}{6}\) represents the total area under the curve from \(x = 0\) to infinity in the first quadrant. As this matches our expectations for asymptotic behavior and convergence, we conclude the solution is valid.

Key concepts

Definite Integral

In integral calculus, a definite integral is used to determine the total accumulation of a quantity, such as area, over an interval. It is represented by the integral sign with upper and lower limits indicating the interval of integration. For example, the definite integral \(\int_{a}^{b} f(x) \, dx\) calculates the area under the curve \(f(x)\) from \(x = a\) to \(x = b\). In our exercise, finding the area between the curve \(y = e^{-6x}\) and the x-axis involves employing a definite integral over the interval \([0, \infty)\). This is a unique case as it involves integrating to infinity, a concept known as an improper integral. To handle such cases, we often substitute a limit approach to find the value of the integral as it progresses towards infinity. This allows us to determine the area despite one of the limits not being finite, translating to the area \(\frac{1}{6}\) units squared in this task.

Exponential Function

The function \(y = e^{-6x}\) is an example of an exponential function where the base \(e\) (approximately 2.718) is raised to a power that is a linear function of \(x\). Exponential functions are powerful tools in mathematics due to their rapid growth or decay properties. In our case, the expression \(e^{-6x}\) signifies an exponential decay, meaning that as \(x\) increases, the value of the function \(y\) decreases.

• Base Functionality: The base \(e\) is constant, and its power dictates the decrease in value.
• Decay Force: The negative exponent \(-6x\) ensures the function decreases or decays.
• Behavior Overview: The curve gets closer to zero but never actually reaches it as \(x\) increases.

Understanding these properties helps make sense of why the function doesn't touch the x-axis right away, illustrating its asymptotic nature, making it perfect for problems concerning indefinite areas like this.

Area Under Curve

When analyzing a curve, one important aspect is calculating the area under it, which is a common task in physics and engineering as it often represents quantities like work done or total accumulated values. Finding the area under a curve involves integrating the function that defines the curve over a specific interval. In our math exercise, the area under \(y = e^{-6x}\) is computed using the definite integral over \(x\) from 0 to infinity.

• Conceptual Interpretation: The area under \(y = e^{-6x}\) represents positive 'space' between the curve and the x-axis.
• Computation Method: Set up an integral, \(\int_{0}^{\infty} e^{-6x} \, dx\), to solve and find the area.
• Result Implication: Result of this integral gives \(\frac{1}{6}\), the total 'positive' area accumulated from zero to infinite.

Thus, computing the area under curves like these gives meaningful physical or conceptual insights about the quantities defined by such functions over a continuum of values.