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Questions and Answers
If $f(x)$ is integrable and $a ≤ c ≤ b$, which property allows you to split the integral $\int_{a}^{b} f(x) dx$ into $\int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$?
If $f(x)$ is integrable and $a ≤ c ≤ b$, which property allows you to split the integral $\int_{a}^{b} f(x) dx$ into $\int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$?
- Constant Multiple
- Sum and Difference
- Reversing Limits of Integration
- Interval Additivity (correct)
Reversing the limits of integration in a definite integral does not affect the sign of the integral.
Reversing the limits of integration in a definite integral does not affect the sign of the integral.
False (B)
What is the value of the definite integral $\int_{a}^{a} f(x) dx$?
What is the value of the definite integral $\int_{a}^{a} f(x) dx$?
0
The integral of a constant multiple of a function, $\int_{a}^{b} k \cdot f(x) dx$, is equal to k multiplied by the integral of the function, which is written as k $\cdot$ $\int_{a}^{b}$ ______ dx.
The integral of a constant multiple of a function, $\int_{a}^{b} k \cdot f(x) dx$, is equal to k multiplied by the integral of the function, which is written as k $\cdot$ $\int_{a}^{b}$ ______ dx.
Match the following integral properties with their descriptions:
Match the following integral properties with their descriptions:
If $f(x)$ is an odd function, what is the value of $\int_{-a}^{a} f(x) dx$?
If $f(x)$ is an odd function, what is the value of $\int_{-a}^{a} f(x) dx$?
If $f(x) \ge g(x)$ for $a ≤ x ≤ b$, then $\int_{a}^{b} f(x) dx$ must be less than or equal to $\int_{a}^{b} g(x) dx$.
If $f(x) \ge g(x)$ for $a ≤ x ≤ b$, then $\int_{a}^{b} f(x) dx$ must be less than or equal to $\int_{a}^{b} g(x) dx$.
According to the comparison property, if $m ≤ f(x) ≤ M$ for $a ≤ x ≤ b$, what inequality can be stated about $\int_{a}^{b} f(x) dx$?
According to the comparison property, if $m ≤ f(x) ≤ M$ for $a ≤ x ≤ b$, what inequality can be stated about $\int_{a}^{b} f(x) dx$?
The absolute value of the definite integral is always ______ than or equal to the definite integral of the absolute value of the function.
The absolute value of the definite integral is always ______ than or equal to the definite integral of the absolute value of the function.
Which property is most useful for simplifying the integral of a piecewise defined function?
Which property is most useful for simplifying the integral of a piecewise defined function?
The integral of an even function $f(x)$ from $-a$ to $a$ is equal to zero.
The integral of an even function $f(x)$ from $-a$ to $a$ is equal to zero.
What does the property $\int_{a}^{b} [f(x) ± g(x)] dx = \int_{a}^{b} f(x) dx ± \int_{a}^{b} g(x) dx$ allow you to do with the integral of a sum or difference of functions?
What does the property $\int_{a}^{b} [f(x) ± g(x)] dx = \int_{a}^{b} f(x) dx ± \int_{a}^{b} g(x) dx$ allow you to do with the integral of a sum or difference of functions?
Which of the following properties is most helpful when estimating bounds on an integral without explicitly solving it?
Which of the following properties is most helpful when estimating bounds on an integral without explicitly solving it?
If $f(x)$ is _____, then $\int{-a}^{a} f(x) dx = 0$.
If $f(x)$ is _____, then $\int{-a}^{a} f(x) dx = 0$.
The absolute value property states that $\int_{a}^{b} f(x) dx \le |\int_{a}^{b} |f(x)| dx|$.
The absolute value property states that $\int_{a}^{b} f(x) dx \le |\int_{a}^{b} |f(x)| dx|$.
How does the integral $\int_{1}^{5} 3f(x) dx$ relate to $\int_{1}^{5} f(x) dx$?
How does the integral $\int_{1}^{5} 3f(x) dx$ relate to $\int_{1}^{5} f(x) dx$?
Explain briefly why $\int_{-5}^{5} x^3 dx = 0$.
Explain briefly why $\int_{-5}^{5} x^3 dx = 0$.
$\int_{3}^{7} f(x) dx + \int_{7}^{9} f(x) dx = \int_{3}$ ______ $f(x) dx$.
$\int_{3}^{7} f(x) dx + \int_{7}^{9} f(x) dx = \int_{3}$ ______ $f(x) dx$.
The absolute value of an integral can be greater than the integral of the absolute value of the function
The absolute value of an integral can be greater than the integral of the absolute value of the function
If you know that $2 ≤ f(x) ≤ 4$ for $1 ≤ x ≤ 6$, what can you say about the value of $\int_{1}^{6} f(x) dx$?
If you know that $2 ≤ f(x) ≤ 4$ for $1 ≤ x ≤ 6$, what can you say about the value of $\int_{1}^{6} f(x) dx$?
Flashcards
Interval Additivity
Interval Additivity
The integral over an interval can be split into integrals over subintervals. ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx, for a ≤ c ≤ b.
Constant Multiple
Constant Multiple
A constant factor within an integral can be factored out: ∫ab k * f(x) dx = k * ∫ab f(x) dx.
Sum and Difference Rule
Sum and Difference Rule
The integral of a sum/difference is the sum/difference of individual integrals: ∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx.
Reversing Limits
Reversing Limits
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Integral of Zero
Integral of Zero
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Symmetric Functions
Symmetric Functions
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Comparison Properties
Comparison Properties
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Absolute Value Property
Absolute Value Property
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Study Notes
- Definite integrals possess properties that facilitate their evaluation and manipulation.
- These properties are fundamental in calculus and are used to simplify complex integrals.
Interval Additivity
- If a function f(x) is integrable on an interval containing a point c, the integral over the entire interval can be split into integrals over subintervals.
- For a ≤ c ≤ b: ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx.
- Useful for piecewise defined functions, where the function's definition changes at certain points within the interval.
- It allows breaking the integral into parts where the function has a consistent definition.
- Also applicable when there's a discontinuity within the interval; can integrate up to the discontinuity and then start again on the other side.
Constant Multiple
- A constant factor within the integrand can be factored out of the integral.
- For any constant k: ∫ab k * f(x) dx = k * ∫ab f(x) dx.
- Simplifies integrand by dealing with constant multipliers separately.
- Useful when the integrand contains a constant that complicates the integration process.
Sum and Difference
- The integral of a sum or difference of functions is the sum or difference of their individual integrals.
- ∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx.
- Facilitates integrating complex expressions by breaking them into simpler terms.
- Applicable when the integrand is a combination of multiple terms that are easier to integrate separately.
Reversing Limits of Integration
- Reversing the limits of integration changes the sign of the integral.
- ∫ba f(x) dx = -∫ab f(x) dx.
- Useful when the limits are in an order that is not conducive to direct integration.
- Can be applied to satisfy certain conditions or to simplify the form of an expression.
Integral of Zero
- The definite integral of a function over an interval of zero length is zero.
- ∫aa f(x) dx = 0.
- If the upper and lower limits of integration are the same, no area is accumulated.
- A direct consequence of the definition of the definite integral.
Integrals of Symmetric Functions
- Symmetric functions (even or odd) have properties that simplify definite integrals over symmetric intervals (e.g., [-a, a]).
- If f(x) is even (f(-x) = f(x)), then ∫-aa f(x) dx = 2 * ∫0a f(x) dx.
- If f(x) is odd (f(-x) = -f(x)), then ∫-aa f(x) dx = 0.
- Exploits symmetry to reduce the computation needed.
- Even functions: the area from -a to 0 is the same as the area from 0 to a.
- Odd functions: the area from -a to 0 is the negative of the area from 0 to a, so they cancel out.
Comparison Properties
- Comparison properties allow us to compare the values of definite integrals based on the integrands.
- If f(x) ≥ 0 for a ≤ x ≤ b, then ∫ab f(x) dx ≥ 0.
- If f(x) ≥ g(x) for a ≤ x ≤ b, then ∫ab f(x) dx ≥ ∫ab g(x) dx.
- If m ≤ f(x) ≤ M for a ≤ x ≤ b, then m(b - a) ≤ ∫ab f(x) dx ≤ M(b - a).
- Useful for estimating bounds on integrals.
- The integral of a non-negative function is non-negative; area under the curve is above the x-axis.
- If one function is greater than another, its integral over the same interval is also greater.
- An integral can be bounded by the product of the interval length and the minimum/maximum value of the function on that interval.
Absolute Value Property
- The absolute value of a definite integral is less than or equal to the definite integral of the absolute value of the function.
- |∫ab f(x) dx| ≤ ∫ab |f(x)| dx.
- Useful when dealing with functions that change sign within the interval of integration.
- Provides an upper bound on the magnitude of the integral.
- The integral of the absolute value represents the total area without regard to sign, which is always greater than or equal to the magnitude of the signed area.
Applying Properties
- Use properties strategically to simplify integrals before attempting to evaluate them directly.
- Look for opportunities to break integrals into smaller intervals, factor out constants, or exploit symmetry.
- Comparison properties are handy when exact evaluation is difficult or unnecessary, and only bounds are required.
- When facing complex integrals, consider each property to see if it can simplify the process.
- Understanding properties can lead to more efficient and accurate solutions.
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