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Chapter 7: Problem 42

Sketch the region and find its area (if the area is finite). \( S = \\{ (x, y) \mid x \le 0, 0 \le y \le e^x \\} \)

Short Answer

Expert verified
The area of the region is 1.

Step by step solution

01

Understand the Given Set

The set \( S \) consists of points \((x, y)\) such that \(x \le 0\) and \(0 \le y \le e^x\). This means that \(x\) is non-positive and \(y\) is between 0 and the exponential function \(e^x\). This restriction of \(y\) implies that the curve \(y = e^x\) serves as the upper boundary of the area for \(x \le 0\).
02

Sketch the Given Set

To sketch the region, first draw the curve \(y = e^x\) which is an exponentially decaying curve for \(x \le 0\). Then, consider the vertical line \(x = 0\), which serves as the right boundary since the inequality is \(x \le 0\). The region below the curve for \(x \le 0\) up to the line \(x = 0\) forms the area of interest.
03

Set Up the Integral for the Area

We need to integrate over the range of \(x\) values from \(x = -\infty\) to \(x = 0\). The formula for the area under a curve from \(a\) to \(b\) for a function \(f(x)\) is given by \(\int_a^b f(x) \, dx\). Here, \(f(x) = e^x\) and \(a = -\infty\), \(b = 0\).
04

Integrate to Find the Area

Set up the integral as \(\int_{-\infty}^{0} e^x \, dx\). The antiderivative of \(e^x\) is \(e^x\). Evaluate this from \(-\infty\) to \(0\).
05

Evaluate the Integral

Calculate the definite integral: \[ \int_{-\infty}^{0} e^x \, dx = \left[ e^x \right]_{-\infty}^{0} = e^0 - \lim_{x \to -\infty} e^x. \] Since \(e^0 = 1\) and \(\lim_{x \to -\infty} e^x = 0\), the area is \(1 - 0 = 1\).

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Key Concepts

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

Exponential Function
An exponential function is a mathematical expression where the variable is in the exponent. A common form is \(f(x) = e^x\), where \(e\) is the mathematical constant approximately equal to 2.71828. This function grows rapidly as \(x\) increases and decays significantly when \(x\) is negative. In our exercise, the curve defined by \(y = e^x\) forms the upper boundary of the region whose area we wish to find. Understanding this function is crucial, as its shape determines the specific region under consideration.
Integration
Integration is a fundamental concept in calculus representing the accumulation of quantities. It's the reverse process of differentiation. When you integrate a function, you're finding the area under its curve within a specified interval. In this problem, integration helps us calculate the area enclosed by the exponential curve and the x-axis to the left of the vertical line \(x = 0\). The process involves setting up an integral that accounts for these boundaries, transforming a complex shape into a manageable numerical value.
Area Calculation
Calculating area under a curve with integration involves using the integral of the function over a certain range. Think of it as slicing the infinite curve into infinitesimally thin rectangles and adding their areas. For the given problem, you take the exponential function \(e^x\) and integrate it over the range from \(x = -\infty\) to \(x = 0\). This tells us the total area under the curve in this region, bounded below by the x-axis. This approach is a staple technique in calculus, essential for determining finite areas even when dealing with infinite bounds.
Definite Integral
A definite integral is a calculated value that represents the exact area under a curve between two bounds, \(a\) and \(b\). For our exercise, we compute the definite integral of \(f(x) = e^x\) from \(x = -\infty\) to \(x = 0\). The integral is expressed as \[ \int_{-\infty}^{0} e^x \, dx. \] By evaluating this, we find the area under \(e^x\) as it slides toward zero. The antiderivative of \(e^x\) is \(e^x\) itself, making the process straightforward. Evaluating the antiderivative from \(-\infty\) yields zero, because the exponential function diminishes to zero as \(x\) approaches \(-\infty\). Therefore, the definite integral calculates the entire finite area, equating to 1 in this scenario.

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Most popular questions from this chapter

(a) Graph the functions \( f(x) = \frac{1}{x^{1.1}} \) and \( g(x) = \frac{1}{x^{0.9}} \) in the viewing rectangles \( [0, 10] \) by \( [0, 1] \) and \( [0, 100] \) by \( [0, 1] \). (b) Find the areas under the graphs of \( f \) and \( g \) from \( x = 1 \) to \( x = t \) and evaluate for \( t = 10, 100, 10^4 , 10^6 , 10^{10} \), and \( 10^{20} \). (c) Find the total area under each curve for \( x \ge 1 \), if it exists.

Find the values of \( p \) for which the integral converges and evaluate the integral for those values of \( p \). \( \displaystyle \int_e^\infty \frac{1}{x (\ln x)^p}\ dx \)

Sketch the region and find its area (if the area is finite). \( S = \\{ (x, y) \mid x \ge 0, 0 \le y \le xe^{-x} \\} \)

(a) Evaluate the integral \( \displaystyle \int_0^\infty e^{-x}\ dx \) for \( n = 0, 1, 2 \), and \( 3 \). (b) Guess the value of \( \displaystyle \int_0^\infty x^n e^{-x}\ dx \) when \( n \) is an arbitrary positive integer. (c) Prove your guess using mathematical induction.

The intensity of light with wavelength \(\lambda\) traveling through a diffraction grating with \(N\) slits at an angle \(\theta\) is given by \(I(\theta)=N^{2} \sin ^{2} k / k^{2}\), where \(k=(\pi N d \sin \theta) / \lambda\) and \(d\) is the distance between adjacent slits. A helium-neon laser with wavelength \(\lambda=632.8 \times 10^{-9} \mathrm{~m}\) is emitting a narrow band of light, given by \(-10^{-6}<\theta<10^{-6}\), through a grating with 10,000 slits spaced \(10^{-4} \mathrm{~m}\) apart. Use the Midpoint Rule with \(n=10\) to estimate the total light intensity \(\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta\) emerging from the grating.
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