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Chapter 5: Problem 7

Evaluate the definite integral. If necessary, review the techniques of integration in your calculus text. $$ \int_{2}^{4} x e^{-x / 2} d x $$

Short Answer

Expert verified
The integral evaluates to \(-10e^{-2} + 6e^{-1}\).

Step by step solution

01

Identify the Integration Technique

The given integral is \( \int_{2}^{4} x e^{-x / 2} \, dx \). This is a product of a polynomial and an exponential function, suggesting the use of integration by parts.
02

Set Up Integration by Parts

The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \). Choose \( u = x \) and \( dv = e^{-x/2} \, dx \). This yields \( du = dx \) and \( v = \int e^{-x/2} \, dx = -2e^{-x/2} \).
03

Apply Integration by Parts Formula

Substitute into the integration by parts formula: \[ \int x e^{-x/2} \, dx = -2x e^{-x/2} \bigg|_{2}^{4} + 2 \int e^{-x/2} \, dx \bigg|_{2}^{4} \].
04

Evaluate the First Term

Calculate \(-2x e^{-x/2} \bigg|_{2}^{4}\): \( [-2 \cdot 4 \cdot e^{-2}] - [-2 \cdot 2 \cdot e^{-1}] \) simplifies to \( -8e^{-2} + 4e^{-1} \).
05

Evaluate the Second Integral

Calculate \(2 \int e^{-x/2} \, dx \bigg|_{2}^{4}\): First, find the indefinite integral as \(-2e^{-x/2}\), then evaluate \([-2e^{-x/2}] \bigg|_{2}^{4} \), which gives \(-2e^{-2} + 2e^{-1}\).
06

Combine Results

Combine results from Steps 4 and 5: \(-8e^{-2} + 4e^{-1} + (-2e^{-2} + 2e^{-1})\). Simplify by combining like terms: \(-10e^{-2} + 6e^{-1}\).
07

Calculate the Numerical Approximation (Optional)

Evaluate \(-10e^{-2} + 6e^{-1}\) using a calculator for practical purposes, obtaining an approximate numerical value.

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Key Concepts

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

Definite Integration
Definite integration is a fundamental concept in calculus. It provides a way to find the accumulation of quantities, such as areas under curves between specific bounds. In the context of the given problem, we are tasked with finding the integral of an expression between the limits 2 and 4. In other words, it measures the area under the curve of the function from x=2 to x=4.

The process involves:
  • Choosing the appropriate integration technique, which in this case is integration by parts.
  • Evaluating the integral over the specified range rather than just finding an indefinite integral.
  • Utilizing the bounds of integration to calculate a specific numerical result.
Definite integrals are widely applicable in fields ranging from physics to economics. They provide insights into real-world problems by allowing us to quantify changes over intervals.
Exponential Functions
Exponential functions are crucial in mathematics due to their unique properties. The expression given in the problem, \( e^{-x/2} \), is an exponential function where \( e \) is the base of natural logarithms, approximately equal to 2.718. These functions grow or decay at rates proportional to their current value, which is why they are often seen in growth and decay processes.

Key features of exponential functions include:
  • Their continuous and smooth curves.
  • A constant base with a variable exponent, allowing them to describe processes that change rapidly or slowly, depending on the sign of the exponent.
  • The derivative and integral of exponential functions are also exponential functions. This makes them uniquely simple to work with in calculus.
In the problem, the exponential function is part of an integrand that also includes a polynomial function, which presents an opportunity to explore integration by parts.
Calculus Techniques
Calculus offers a variety of techniques for solving integrals and derivatives. In this particular exercise, we focus on integration by parts, a valuable integration technique useful when dealing with the product of two different types of functions.

The steps for integration by parts are:
  • Identify the parts of the integral \( u \) and \( dv \) such that \( u \) can be easily differentiated and \( dv \) can be easily integrated.
  • Apply the formula \( \int u \, dv = uv - \int v \, du \).
  • Evaluate the resulting simpler integrals and combine the results.
This technique is particularly handy when polynomial and exponential functions are multiplied together. By selecting the polynomial part as \( u \) and the exponential part as \( dv \), the resulting integrals tend to simplify the problem.

Understanding and applying these techniques correctly can significantly boost problem-solving efficiency in calculus, especially with seemingly complex expressions.

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Most popular questions from this chapter

Evaluate the given integral along the indicated contour. \(\int_{C}\left(x^{2}+i y^{3}\right) d z\), where \(C\) is the straight line from \(z=1\) to \(z=i\)

The flow described by the velocity field \(f(z)=(a+i b) / \bar{z}\) is said to have a vortex at \(z=0 .\) The geometric nature of the streamlines depends on the choice of \(a\) and \(b\). (a) Show that if \(z(t)=x(t)+i y(t)\) is the path of a particle in the flow, then $$ \begin{aligned} &\frac{d x}{d t}=\frac{a x-b y}{x^{2}+y^{2}} \\ &\frac{d y}{d t}=\frac{b x+a y}{x^{2}+y^{2}} \end{aligned} $$ (b) Rectangular and polar coordinates are related by \(r^{2}=x^{2}+y^{2}\), tan \(\theta=y / x\). Use these equations to show that $$ \frac{d r}{d t}-\frac{1}{r}\left(x \frac{d x}{d t}+y \frac{d y}{d t}\right), \frac{d \theta}{d t}-\frac{1}{r^{2}}\left(-y \frac{d x}{d t}+x \frac{d y}{d t}\right) $$ (c) Use the equations in parts (a) and (b) to establish that $$ \frac{d r}{d t}=\frac{a}{r}, \frac{d \theta}{d t}=\frac{b}{r^{2}} $$ (d) Use the equations in part (c) to conclude that the streamlines of the flow are logarithmic spirals \(r=c e^{a \theta / b}, b \neq 0 .\) Use a graphing utility to verify that a particle traverses a path in a counterclockwise direction if and only if \(a<0\), and in a clockwise direction if and only if \(b<0 .\) Which of these directions corresponds to motion spiraling into the vortex?

Evaluate \(\oint_{C}\left(x^{2}-y^{2}\right) d s\), where \(C\) is given by \(x=5 \cos t, y=5 \sin t, 0 \leq t \leq 2 \pi\).

In Problems 17 and 18 , the given analytic function \(\boldsymbol{\Omega}(z)\) is a complex velocity potential for the flow of an ideal fluid. Find the velocity field \(\mathbf{F}(x, y)\) of the flow. $$ \Omega(z)=\frac{1}{3} i z^{3} $$

Find an upper bound for the absolute value of the integral \(\int_{C} \operatorname{Ln}(z+3) d z\) where the contour \(C\) is the line segment from \(z=3 i\) to \(z=4+3 i\).
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