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Questions and Answers
What is another name for indefinite integration?
What is another name for indefinite integration?
Which type of integration involves finding the exact numerical value of the area under a curve between two points?
Which type of integration involves finding the exact numerical value of the area under a curve between two points?
What does the definite integration process involve besides finding areas under curves?
What does the definite integration process involve besides finding areas under curves?
In which type of integration do we produce families of functions instead of just one value?
In which type of integration do we produce families of functions instead of just one value?
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Which rule of indefinite integration states that if two functions have the same derivative, their difference is a constant?
Which rule of indefinite integration states that if two functions have the same derivative, their difference is a constant?
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Which type of integration involves the process of finding a function whose derivative matches the given function?
Which type of integration involves the process of finding a function whose derivative matches the given function?
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What does the definite integral between a and b, denoted by $\int_{a}^{b} f(x),dx$, represent?
What does the definite integral between a and b, denoted by $\int_{a}^{b} f(x),dx$, represent?
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What happens when you reverse the order of limits in a definite integral?
What happens when you reverse the order of limits in a definite integral?
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When is the integral of a function from one endpoint to a crossing point positive?
When is the integral of a function from one endpoint to a crossing point positive?
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Which property holds true for additivity of definite integrals?
Which property holds true for additivity of definite integrals?
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Integration by parts involves breaking down an integral into how many simpler integrals?
Integration by parts involves breaking down an integral into how many simpler integrals?
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In integration by parts, if f(x) = u(x)v(x), then what does F(x) = ∫f(x) dx represent?
In integration by parts, if f(x) = u(x)v(x), then what does F(x) = ∫f(x) dx represent?
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Study Notes
Integration
Integration is a fundamental concept in calculus and is used to find areas under curves, determine volumes of solids, and solve problems involving accumulation rates. There are two main types of integration:
Indefinite Integration
Indefinite integration, also known as antidifferentiation, is the process of finding a function whose derivative matches the given function. It produces families of functions rather than just one value due to the presence of arbitrary constants. For example, the integral of f(x) = x² with respect to x gives F(x) = (\dfrac{1}{3}x^3), where k is the constant of integration.
Rules of Indefinite Integration
The rules of indefinite integration include integrating with respect to x, multiplying by constants, adding and subtracting integrals, and using substitution. Some common examples include:
- Integral of (\int u , du = u + C).
- (\int x^n dx = \dfrac{x^{n+1}}{n+1} + C).
- If (F'(x)) and (G'(x)) are both equal to some function (f(x)), and (C_1) and (C_2) are constants, then (F(x)) and (G(x)) differ by a constant, i.e., (F(x) - G(x) = C_1 - C_2).
- To integrate a trigonometric function, we may use the power rule, product rule, quotient rule, chain rule, and other rules.
Definite Integration
Definite integration, also called definite summation or the Riemann sum, involves finding the exact numerical value of the area under a curve between two points. This method uses the Fundamental Theorem of Calculus and has applications in physics and engineering. In general, the definite integral between a and b is denoted as [\int_{a}^{b} f(x),dx.]
Properties of Definite Integrals
Properties of definite integrals include the following rules:
- Reversing the order of limits gives you the negative of the original result, since the reverse order corresponds to the opposite direction of the interval.
- The definite integral can be divided into intervals.
- Additivity holds between any two different intervals. That is, if [A = \int_{a}^{a+h} f(x) , dx] and [B = \int_{a+h}^{b} f(x) , dx,] then A + B = B - A.
- If the graph of f crosses the x axis, then the integral from the intersection point to either end of the interval will be positive. Consequently, the difference between the integrals from each endpoint to a crossing point will give the signed area above the axis.
Integration by Parts
Integration by parts is another technique used to evaluate definite integrals. It involves breaking down the integral into two simpler integrals that can be integrated separately. Let f (x) = u(x)v(x) be a product of two functions and let F(x) = ∫f(x) dx and G(x)= ∫v(x) dx. Then, the Fundamental Theorem of Calculus tells us that F(x) = u(x)G(x) + C. Therefore, the integral of the product of two functions can be found using this formula.
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Description
Explore the fundamental concepts of integration in calculus, including indefinite integration (antidifferentiation), definite integration (definite summation), and integration by parts. Learn about rules and properties for both indefinite and definite integrals, as well as techniques for evaluating integrals of various functions.
