Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 21
In Exercises \(15-22,\) graph the integrands and use known area formulas to evaluate the integrals. $$ \int_{-1}^{1}(2-|x|) d x $$
Short Answer
Expert verified
The integral evaluates to 1.
Step by step solution
01
Understanding the integrand
The given integrand is a function, defined as \(2 - |x|\). This function is a piecewise linear function that forms a V-shape when graphed. As we know, \(|x|\) is the absolute value of \(x\), which means this function measures the distance of \(x\) from 0 without considering the direction.
02
Graph the integrand function
The function \(2 - |x|\) can be split into two linear equations based on the absolute value:- For \(x \geq 0\), the function is \(f(x) = 2 - x\).- For \(x < 0\), the function is \(f(x) = 2 + x\). The graph is a V-shaped figure with a vertex at \((0, 2)\) and intercepts at \((1, 1)\) and \((-1, 1)\).
03
Identify the geometry of the graph
The graph of \(2 - |x|\) between \(-1\) and \(1\) forms an isosceles triangle. The vertices of this triangle are \((-1, 1)\), \((1, 1)\), and the vertex \((0, 2)\).
04
Calculate the base and height of the triangle
The base of the triangle extends horizontally from \(-1\) to \(1\), giving a base length of \(2\). The height of the triangle extends vertically from the base point (0, 1) up to the vertex (0, 2), giving a height of \(1\).
05
Apply the area formula for a triangle
To find the area of the triangle, we use the area formula for a triangle: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\).Substitute the base and height values: \[\text{Area} = \frac{1}{2} \times 2 \times 1 = 1.\]
06
Evaluate the integral
Since the area under the curve \(2 - |x|\) from \(-1\) to \(1\) corresponds to the area of the triangle, the integral \(\int_{-1}^{1}(2-|x|) dx\) evaluates to the area of the triangle, which is \(1\).
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.
Piecewise Linear Functions
A piecewise linear function is a type of function built from straight-line sections. It's like connecting dots on a graph with straight lines. Just like in a puzzle, each segment or piece fits a specific part of the graph based on the input value (in this case, \(x\)).
In our exercise, the function \(2 - |x|\) is piecewise linear because it behaves differently depending on whether \(x\) is positive or negative.
In our exercise, the function \(2 - |x|\) is piecewise linear because it behaves differently depending on whether \(x\) is positive or negative.
- For \(x \geq 0\), the function is \(2 - x\), which descends linearly as \(x\) increases.
- For \(x < 0\), the function is \(2 + x\), which ascends linearly as \(x\) decreases.
Absolute Value
The absolute value function is a mathematical concept that determines the distance a number is from zero on a number line, without regard to direction. In simpler terms, it makes any number positive.
In the formula \(|x|\), this symbol "| |" indicates absolute value, taking any given \(x\) and converting it into its non-negative counterpart.
In the formula \(|x|\), this symbol "| |" indicates absolute value, taking any given \(x\) and converting it into its non-negative counterpart.
- If \(x = 3\), then \(|x| = 3\).
- If \(x = -5\), then \(|x| = 5\).
Area Under a Curve
Finding the area under a curve is a fundamental concept in calculus, closely related to definite integrals. It involves calculating the space between the curve of a function and the x-axis over a specified interval.
This concept is like measuring the amount of paint needed to cover a shape bounded by the curve and the axis.
This concept is like measuring the amount of paint needed to cover a shape bounded by the curve and the axis.
- The graph of \(2 - |x|\) from \(-1\) to \(1\) forms a triangular shape.
- The area under this curve can be found using geometry because the shape is a triangle.
Triangle Area Formula
The area of a triangle can be computed using the simple formula: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). This equation helps determine the extent of a two-dimensional space enclosed by a triangular figure.
- The base is one side of the triangle, often parallel to the x-axis in graph problems.
- The height is the perpendicular distance from the base to the opposite vertex.
