Chapter 5: Problem 35
Use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x .\) Begin by drawing a graph of \(f\). $$f(x)=|x-2|$$
Short Answer
Expert verified
The integral evaluates to 4.
Step by step solution
01
Graph the Function
First, graph the function \( f(x) = |x - 2| \). It is a V-shaped graph with a vertex at \( x = 2 \). The function is linear on both sides of the vertex: it decreases as a diagonal line from \( x = 0 \) to \( x = 2 \), and increases from \( x = 2 \) to \( x = 4 \).
02
Break Down the Integral
Use the Interval Additive Property to split the integral \( \int_{0}^{4} f(x) \, dx \) into two parts: \( \int_{0}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx \). This takes advantage of the linearity where the function changes slope.
03
Evaluate the First Integral
For \( \int_{0}^{2} f(x) \, dx \), substitute \( f(x) = |x-2| = 2-x \). The integral becomes \( \int_{0}^{2} (2-x) \, dx \). This evaluates to \( [2x - \frac{x^2}{2}]_{0}^{2} = (4 - 2) - (0 - 0) = 2 \).
04
Evaluate the Second Integral
For \( \int_{2}^{4} f(x) \, dx \), substitute \( f(x) = |x-2| = x-2 \). The integral becomes \( \int_{2}^{4} (x-2) \, dx \). This evaluates to \( [\frac{x^2}{2} - 2x]_{2}^{4} = (8 - 8) - (2 - 4) = 2 \).
05
Combine the Results
Add the results of the two integrals: \( 2 + 2 \) which gives \( 4 \). Therefore, \( \int_{0}^{4} f(x) \, dx = 4 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Additive Property
In calculus, the Interval Additive Property is an important concept when evaluating definite integrals over multiple intervals.
This property states that if you have a continuous function over an interval \[a,b\], you can split the integral into smaller intervals, add them together, and get back the whole integral.
For instance, if we're looking at \(\int_{0}^{4} f(x) \, dx\), we can break it into \[\int_{0}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx\]. This takes advantage of the function's behavior across the entire range by grouping certain properties within tightened bounds.
This is especially useful if the function has different behaviors or simpler forms within the smaller intervals, as seen in the piecewise nature of many absolute value functions. Thus, the total area under the graph of the function is exactly the sum of the areas from individual segments.
This property states that if you have a continuous function over an interval \[a,b\], you can split the integral into smaller intervals, add them together, and get back the whole integral.
For instance, if we're looking at \(\int_{0}^{4} f(x) \, dx\), we can break it into \[\int_{0}^{2} f(x) \, dx + \int_{2}^{4} f(x) \, dx\]. This takes advantage of the function's behavior across the entire range by grouping certain properties within tightened bounds.
This is especially useful if the function has different behaviors or simpler forms within the smaller intervals, as seen in the piecewise nature of many absolute value functions. Thus, the total area under the graph of the function is exactly the sum of the areas from individual segments.
Linearity of Integration
Linearity of integration is based on the principle that integration obeys both additive and scalar multiplication rules, simplifying the process of integrating complex functions.
When you integrate, \[\int (af(x) + bg(y)) \,dx = a\int f(x) \,dx + b\int g(y) \,dy\], where \(a\) and \(b\) are constants.
This rule makes it possible to evaluate definite integrals by separating and resolving each part independently and then combining them. For example, the integral \(\int (2-x) \, dx\) simplifies the process by allowing you to deal with each term's integral separately.
The term "linearity" stems from how it maintains straight-line combinations in function evaluation, maintaining simplicity even when dealing with functions involving sums or differences.
When you integrate, \[\int (af(x) + bg(y)) \,dx = a\int f(x) \,dx + b\int g(y) \,dy\], where \(a\) and \(b\) are constants.
This rule makes it possible to evaluate definite integrals by separating and resolving each part independently and then combining them. For example, the integral \(\int (2-x) \, dx\) simplifies the process by allowing you to deal with each term's integral separately.
The term "linearity" stems from how it maintains straight-line combinations in function evaluation, maintaining simplicity even when dealing with functions involving sums or differences.
Absolute Value Function
The absolute value function, denoted as \(|x|\), is a piecewise-defined function that returns the distance of a number from zero on the real number line.
The expression \(|x-2|\) shifts this function horizontally by two units. \(|x-2|\) equals \(x-2\) when \(x\) is greater than or equal to 2, and \(2-x\) when \(x\) is less than 2.
It's visualized typically as a "V" shape on a graph where the vertex is at the point where the inner expression equals zero. Recognizing the breakpoints of an absolute value function is crucial when setting up integrals, as it determines where the function changes its definition, and therefore, how these integrals should be evaluated across intervals.
The expression \(|x-2|\) shifts this function horizontally by two units. \(|x-2|\) equals \(x-2\) when \(x\) is greater than or equal to 2, and \(2-x\) when \(x\) is less than 2.
It's visualized typically as a "V" shape on a graph where the vertex is at the point where the inner expression equals zero. Recognizing the breakpoints of an absolute value function is crucial when setting up integrals, as it determines where the function changes its definition, and therefore, how these integrals should be evaluated across intervals.
Piecewise Functions
Piecewise functions are those defined by different expressions over different intervals of the domain. They are particularly useful in modeling situations where a function behaves differently depending on the input value.
For example, the function \(f(x) = |x - 2|\) is piecewise because it has one rule \(f(x) = 2-x\) for \(x < 2\) and another rule \(f(x) = x-2\) for \(x \geq 2\).
Evaluating piecewise functions involves identifying the intervals and evaluating each part separately. In integration, this might mean splitting the integral at the point where the function changes form. This way, each part can be integrated separately based on its specific behavior over its defined interval, ensuring accurate results. Recognizing and using piecewise functions ensures that varying behaviors in real-world applications can be accurately captured and processed mathematically.
For example, the function \(f(x) = |x - 2|\) is piecewise because it has one rule \(f(x) = 2-x\) for \(x < 2\) and another rule \(f(x) = x-2\) for \(x \geq 2\).
Evaluating piecewise functions involves identifying the intervals and evaluating each part separately. In integration, this might mean splitting the integral at the point where the function changes form. This way, each part can be integrated separately based on its specific behavior over its defined interval, ensuring accurate results. Recognizing and using piecewise functions ensures that varying behaviors in real-world applications can be accurately captured and processed mathematically.
