Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 19
Sketch the integrand of the given definite integral over the interval of integration. Evaluate the integral by calculating the area it represents. $$ \int_{-1}^{3}|x| d x $$
Short Answer
Expert verified
The integral \( \int_{-1}^{3} |x| \, dx \) evaluates to 5.
Step by step solution
01
Understanding the Integrand
The integrand is the absolute value function, \(|x|\), which takes the value \(x\) when \(x \geq 0\) and \(-x\) when \(x < 0\). This creates a V-shaped graph that is symmetric about the y-axis.
02
Sketching the Graph
Draw the graph of \(|x|\) over the interval \([-1, 3]\). It will be a V-shaped graph with the point at the origin (0,0). From \(-1\) to 0, the graph is increasing, and from 0 to 3, it continues with a positive slope line.
03
Identifying Areas Under the Sections
The area needed is split into two regions: one from \([-1,0]\) and another from \([0,3]\). Compute these areas to evaluate the integral.
04
Calculating Area from -1 to 0
In the interval \([-1,0]\), the function \(|x| = -x\). The area under this line is the area of a right triangle with a base of 1 and height of 1: \( \text{Area} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}\).
05
Calculating Area from 0 to 3
In the interval \([0,3]\), the function \(|x| = x\). The area under this line is the area of a right triangle with a base of 3 and height of 3: \( \text{Area} = \frac{1}{2} \times 3 \times 3 = \frac{9}{2}\).
06
Adding Areas from Both Sections
Add the areas computed in the previous steps. The total area under the curve, representing the integral, is \( \frac{1}{2} + \frac{9}{2} = 5 \).
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.
Absolute Value Function
The absolute value function, denoted as \(|x|\), is a fundamental mathematical function that measures the distance of a number from zero on the number line. Essentially, it converts negative numbers into positive ones while leaving positive numbers unaffected.
- Mathematically, it is defined as \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\).
- Graphically, the plot of \(|x|\) results in a V-shaped curve that is symmetric about the y-axis.
- This V-shape is due to the piecewise nature of the function, transitioning smoothly from one piece to another at the origin.
Area Under a Curve
Evaluating a definite integral often involves finding the area under a curve on a graph. For the absolute value function, this concept is visually interpreted as the sum of areas under the line segments that form the V-shape.
- The process begins with identifying the intervals where different expressions of the function apply.
- In this problem, the function \(|x|\) has two segments on the interval \([-1,3]\); one for \([-1,0]\) and another for \([0,3]\).
- For \([-1,0]\), the area of the triangle is computed using \(\frac{1}{2} \times \text{{base}} \times \text{{height}} \).
- From \([0,3]\), the function simplifies to a similar calculation for another triangular area.
Graph Sketching
Sketching a graph is a practical way to visualize functions and their properties. For the function \(|x|\), sketching the graph helps in clearly identifying the areas under the curve.
- The graph for \(|x|\) over an interval \([-1,3]\) reflects a symmetrical V-shape centered at the origin.
- From points \([-1,0]\), the graph ascends linearly with a negative slope as \(|x| = -x\).
- From \(0\) onward to \(3\), it continues with a positive slope where \(|x| = x\).
- Make sure to accurately plot the vertex at the origin and properly scale axis marks.
- Mark intervals correctly to ensure accurate calculations of areas under curves.
Piecewise Function
The absolute value function serves as a quintessential example of a piecewise function. In mathematics, piecewise functions are defined by multiple sub-functions, each applying to a certain interval or portion of the domain.
- In the case of \(|x|\), there are two pieces: \(|x| = -x\) for negative x-values and \(|x| = x\) for non-negative x-values.
- This division creates distinct linear segments with different slopes, explaining the V-shape.
- Analyze each segment separately for its defined interval, focusing on where and how the function switches between pieces.
- For integrals, this segmentation aids in determining which formula to apply over specific intervals.
