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Chapter 4: Problem 11

Find the area under the given curve over the indicated interval. $$y=e^{x}:[0,3]$$

Short Answer

Expert verified
The area under the curve \(y = e^{x} \) from \([0, 3]\) is \[ e^{3} - 1 \].

Step by step solution

01

- Identify the Integral

The area under a curve from point A to point B is given by the definite integral of the function from A to B. Here, we want to find the area under the curve described by the function \(y = e^{x}\) over the interval \([0, 3]\). So, we need to compute \(\begin{equation*} \text{Area} = \int_{0}^{3} e^{x} \, dx \end{equation*}\).
02

- Compute the Antiderivative

The next step is to find the antiderivative (or indefinite integral) of \(e^{x}\). The antiderivative of \(e^{x}\) is \(e^{x}\).
03

- Evaluate the Definite Integral

Using the Fundamental Theorem of Calculus, evaluate the antiderivative from 0 to 3:\begin{equation*} \text{Area} = \[ e^{x} \]_{0}^{3} = e^{3} - e^{0} \end{equation*}. Recall that \ e^{0} = 1\.
04

- Simplify the Result

Finally, simplify the result from the previous step. So, the area is \[ e^{3} - 1 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integral
To find the area under a curve within a specific interval, we use a method called a definite integral. A definite integral calculates the total accumulation of quantities, providing a precise area under the curve. In this case, the exercise asks for the area under the curve described by the exponential function \(y = e^{x}\) from \[0, 3\]. Here's how to understand it:
  • Identify the interval: \[0, 3\]
  • Set up the integral: \(\begin{equation*} \text{Area} = \int_{0}^{3} e^{x} \, dx \end{equation*}\)
  • Use the properties of definite integrals: Area represents the total accumulation between the x-values 0 and 3.
Each of these steps is crucial for solving the problem systematically.
Antiderivative
In calculus, finding an antiderivative (or indefinite integral) of a function is akin to reversing differentiation. For the function \(e^{x}\), finding the antiderivative is straightforward. The antiderivative of \(e^{x}\) is simply \(e^{x}\).
Here's why this is important:
  • An antiderivative provides a function whose derivative is the original function.
  • It allows us to evaluate the definite integral with the Fundamental Theorem of Calculus: \[ F(b) - F(a) \], where \(F\) is the antiderivative.
Specifically, for \(y = e^{x}\), we identify \(F(x) = e^{x}\). This step simplifies the process of finding the specific value of the integral from 0 to 3.
Exponential Function
An exponential function, like \(e^{x}\), is unique because it grows exponentially, meaning its rate of growth increases over time. This makes it a powerful function in many mathematical applications. What you need to remember about exponential functions:
  • The function \(e^{x}\) is the exponential function with base \(e\), where \(e\) is approximately equal to 2.718.
  • The derivative and antiderivative of \(e^{x}\) are both \(e^{x}\), highlighting its distinct properties.
In this context, knowing the properties of \(e^{x}\) helps us find the specific area under the curve. The final evaluation involves computing \(e^{3} - e^{0} = e^{3} - 1\), simplifying it to a clear result.

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