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Magazine Chapter 6: Problem 21
Find the area enclosed by the given curves. $$ y=e^{|x|}, y=0, x=-1, x=2 $$
Short Answer
Expert verified
The area enclosed is \(e^{2} + e^{-1} - 2\).
Step by step solution
01
Understand the Problem
We need to find the area between the curve defined by \(y = e^{|x|}\), the x-axis \(y = 0\), and the vertical lines \(x = -1\) and \(x = 2\). We will calculate the definite integral over this interval.
02
Analyze the Function
The function \(y = e^{|x|}\) can be split into two cases due to the absolute value: \(y = e^{-x}\) for \(x \leq 0\) and \(y = e^{x}\) for \(x \geq 0\). Since \(-1 \leq x \leq 2\), we need to consider both cases.
03
Calculate the Integral for \(x < 0\)
For \(x = -1\) to \(x = 0\), the expression is \(y = e^{-x}\). The area is given by the integral \( \int_{-1}^{0} e^{-x} \, dx \).
04
Solve the Integral for \(x < 0\)
Calculate the integral: \[ \int e^{-x} \, dx = -e^{-x} + C \] Evaluating from -1 to 0:\[-[e^{-0} - e^{-(-1)}] = -(1 - e^{-1}) = e^{-1} - 1.\]
05
Calculate the Integral for \(x > 0\)
For \(x = 0\) to \(x = 2\), the expression is \(y = e^{x}\). The area is given by the integral \( \int_{0}^{2} e^{x} \, dx \).
06
Solve the Integral for \(x > 0\)
Calculate the integral:\[ \int e^{x} \, dx = e^{x} + C \]Evaluating from 0 to 2:\[e^{2} - e^{0} = e^{2} - 1.\]
07
Sum the Areas
Add the areas from both parts of the integral to find the total area:\[e^{-1} - 1 + e^{2} - 1 = e^{2} + e^{-1} - 2.\]
08
Finalize and Verify
The total area between the curves is verified by adding calculated parts. Ensure correct syntax and calculations to finalize.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
The definite integral is a powerful mathematical tool used to calculate the area under a curve bounded by specific limits. In this context, we aim to find the area between the curve described by the function and the x-axis, from a starting point to an ending point. It provides a numerical value representing this area.
- The definite integral is expressed as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the bounds of integration.
- The function \(f(x)\) is the curve we are interested in.
- The result represents the net area between the curve \(f(x)\) and the x-axis from \(x = a\) to \(x = b\).
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. A classic example is \(y = e^{x}\), where \(e\) is a mathematical constant approximately equal to 2.71828. They have significant properties such as growth rates and continuous compounding.
- In exponential functions, the variable \(x\) appears in the exponent position.
- The function \(y = e^{|x|}\) used in our problem showcases exponential growth and decay, depending on the value of \(x\).
- For positive \(x\), the function grows, \(e^{x}\), and for negative \(x\), it represents decay, \(e^{-x}\).
Piecewise Functions
Piecewise functions are defined by different expressions depending on the input value. This characteristic allows the function to exhibit different behaviors over various intervals. Our exercise involves a classic piecewise function due to the presence of an absolute value.
- The function \(y = e^{|x|}\) behaves differently based on whether \(x\) is positive, negative, or zero.
- In our case, it splits into two distinct expressions: \(y = e^{-x}\) for \(x \leq 0\) and \(y = e^{x}\) for \(x \geq 0\).
- This causes the need to compute two separate integrals: one for each distinct behavior of the function.
