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By using the fundamental theorem of Calculus. i.e. The integral of f(x) = F(x), your limits are [a,b]. Solve: F(b) - F(a). The FTC, second part, says that if f is a continuous real valued function of [a,b] then the integral from a to b of f(x)= F(b) - F(a) where F is any antiderivative of f, that is, a function such that F'(x) = f(x).
Example: Evaluate the integral form -2 to 3 of x^2. The integral form -2 to 3 of x^2 = F(-2) - F(3) = -2^3/3 - 3^3/3 = -8/3 - 27/3 = -35/3
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What is integral of 2x?
The integral of 2x is x^2+c, where c is a constant. If this is a definite integral, meaning that the limits of integration are known, then c=0. If this is an indefinite integral, meaning the limits of integration are unknown, then c should either be left as is or solved for using an initial condition.
Application of definite Integral in the real life Give some?
What are the Applications of definite integrals in the real life?
How do you integrate 4 over 3-x?
4 is a constant, so you can pull it out of the integral. Use a u substitution with u = 3-x and du = -dx. If it's a definite integral, remember to change the limits of integration. The integral is then simply 1/u which integrates to be ln u. Substitute back in 3-x for u and you have the answer to be: -4*ln (3-x) + C
How do you evaluate definite integrals?
In order to evaluate a definite integral first find the indefinite integral. Then subtract the integral evaluated at the bottom number (usually the left endpoint) from the integral evaluated at the top number (usually the right endpoint). For example, if I wanted the integral of x from 1 to 2 (written with 1 on the bottom and 2 on the top) I would first evaluate the integral: the integral of x is (x^2)/2 Then I would subtract the integral evaluated at 1 from the integral evaluated at 2: (2^2)/2-(1^2)/2 = 2-1/2 =3/2.
What are some real life applications of indefinite integrals?
One of the major applications of indefinite integrals is to calculate definite integrals. If you can't find the indefinite integral (or "antiderivative") of a function, some sort of numerical method has to be used to calculate the definite integral. This might be seen as clumsy and inelegant, but it is often the only way to solve such a problem.Definite integrals, in turn, are used to calculate areas, volumes, work, and many other physical quantities that can be expressed as the area under a curve.
