Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 47
Evaluate the definite integral. Use a graphing utility to confirm your result. $$ \int_{0}^{4} x e^{-x / 2} d x $$
Short Answer
Expert verified
The evaluated integral is approximately 1.2094
Step by step solution
01
Apply Integration by Parts
In integration by parts, the integral of a product of two functions is given by \(uv - \int v du\). By letting \(u = x\) and \(dv = e^{-x / 2} dx\), we can find that \(du = dx\) and \(v = -2e^{-x / 2}\).
02
Apply the Integration by Parts Formula
Applying these values to the formula, the integral becomes: \(-2x e^{-x / 2} - \int -2 e^{-x / 2} dx\). This simplifies further to \(-2x e^{-x / 2} + 4e^{-x / 2} \).
03
Evaluate the Definite Integral
We substitute the limits 0 and 4 into the integral to find its numerical value: \( -2x e^{-x / 2} + 4e^{-x / 2} \bigg|_0^4 \). This simplifies to \( -8e^{-2} + 4e^{-2} - 4 \).
04
Compute Value
By calculating the values, we can simplify the result to 4 - 16 e^{-2} approximately equal to 1.2094.
05
Confirm the Result with a Graphing Utility
To confirm the result visually, plot the function \( f(x) = xe^{-x/2} \) on a graphing utility and calculate integral from 0 to 4, which should give a close value to the calculated result.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration by Parts
Integration by parts is a valuable technique in calculus for evaluating integrals, especially those involving a product of functions. The general formula you use is \[ \int u \, dv = uv - \int v \, du \]. Here鈥檚 how it works:
- Selecting functions: Choose which part of the integrand is \(u\) and which is \(dv\). A common rule is to let \(u\) be a function that becomes simpler when differentiated, while \(dv\) is something easily integrated.
- Computing derivatives and integrals: Once you have \(u\) and \(dv\), find \(du\) by differentiating \(u\), and \(v\) by integrating \(dv\).
- Substituting into the formula: Substitute \(u\), \(v\), \(du\) into the integration by parts formula to transform the original integral into a hopefully simpler one.
- \(u = x\), hence \(du = dx\).
- \(dv = e^{-x/2} \, dx\), resulting in \(v = -2e^{-x/2}\).
Graphing Utility
A graphing utility is a digital tool that helps visualize functions and confirm integrals or other calculus problems. It can be an application on a calculator or a computer-based software such as Desmos or GeoGebra. Here鈥檚 how you can make the most of it:
- Visualizing Functions: Plotting allows you to see the function鈥檚 behavior and its curve. It offers a geometric understanding which might not be obvious from the equation alone.
- Evaluating Integrals: After you鈥檝e calculated an integral by hand, using a graphing utility provides confirmation. Input the function and set the interval, and the utility will compute the area under the curve.
- Checking Work: If your integral鈥檚 result from the utility closely matches your hand-calculated value, it reassures the work is likely correct.
Numerical Integration
Numerical integration is a strategy to approximate the value of a definite integral. It's especially useful when the integral does not have an elementary antiderivative or is complex. Popular techniques include:
Though for our integral \(\int_{0}^{4} xe^{-x/2} \, dx\), we solved it analytically, numerical methods can be excellent for confirming results or handling functions that can't easily be integrated by basic calculus methods.
- Trapezoidal Rule: Approximates the area under a curve by dividing the total area into trapezoids rather than rectangles. It's a step up in accuracy from simpler methods like Riemann sums.
- Simpson鈥檚 Rule: This combines parabolic arcs rather than straight lines to better fit the curve being integrated. It often yields more accurate results than the trapezoidal rule.
Though for our integral \(\int_{0}^{4} xe^{-x/2} \, dx\), we solved it analytically, numerical methods can be excellent for confirming results or handling functions that can't easily be integrated by basic calculus methods.
