Integral calculus is a branch of calculus that deals with finding the antiderivative of a function and calculating the area under a curve. It involves techniques such as integration by substitution, integration by parts, and partial fractions. The fundamental theorem of calculus states that the definite integral of a function can be found by evaluating its antiderivative at the endpoints of the interval and subtracting the results. Integral calculus is used in a wide range of fields, including physics, engineering, economics, and statistics.
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Google Gemini AI
May 18
Integral Calculus
Integral calculus is a branch of mathematics that deals with the process of finding the area under a curve, the volume of a solid, and the length of a curve. It is the inverse operation of differential calculus, which deals with the process of finding the derivative of a function.
Fundamental Concepts:
- Integral: The integral of a function f(x) over an interval [a, b] is the area under the curve y = f(x) from a to b. It is denoted as:
``` ∫[a,b] f(x) dx ```
- Antiderivative: An antiderivative of a function f(x) is a function g(x) whose derivative is f(x). It is denoted as:
``` ∫f(x) dx = g(x) + C ```
where C is an arbitrary constant.
- Definite Integral: The definite integral of a function f(x) over an interval [a, b] is the net area under the curve y = f(x) from a to b. It is denoted as:
``` ∫[a,b] f(x) dx ```
and is a number.
- Indefinite Integral: The indefinite integral of a function f(x) is the set of all antiderivatives of f(x). It is denoted as:
``` ∫f(x) dx ```
Applications:
Integral calculus has numerous applications in various fields, including:
- Area and Volume: Calculating the area under a curve or the volume of a solid. - Length of Curves: Finding the length of a curved line. - Work and Energy: Calculating the work done by a force or the energy of a system. - Statistics: Estimating probabilities and finding moments of distributions. - Physics: Solving problems involving motion, fluid dynamics, and heat transfer.
Methods of Integration:
- Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C - Substitution Rule: ∫f(g(x)) g'(x) dx - Integration by Parts: ∫u dv = uv - ∫v du - Trigonometric Integrals: ∫sin(x) dx = -cos(x) + C, ∫cos(x) dx = sin(x) + C - Logarithmic Integrals: ∫1/x dx = ln|x| + C
Chain Rule of Integration:
If f(x) is a composite function, then the integral of f(x) is: