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Discuss the role of constant of integration in indefinite integrals, why we don't involve it in definite integrals? (Be precise)
Opening Date: Feb 6, 2013 at 12:01 AM
Closing Date: Feb 7, 2013 at 11:59 PM
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Idea..
You can approximate the area under a curve by adding up right, left, or midpoint rectangles. To find an exact area, you need to use a definite integral.
When you approximate the area under a curve, the tops of the rectangles form a saw tooth shape that doesn’t fit perfectly along the smooth curving function. So, to find the exact area under a curve using rectangles, you’d need to find the area of an infinite number of infinitely thin rectangles whose “tops” do perfectly fit the curve. Now, you can’t really use an infinite number of rectangles, but with the fantastic invention of limits, this is sort of what happens.
Here’s the “simple” definition of the definite integral that’s used to compute exact areas. It’s based on the limit of a Riemann sum of right rectangles. The exact area under a curve between a and b is given by the definite integral, which is defined as follows:
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In the equation
the letter C is called the constant of integration. The function f ( x ) is still called the integrand, like it was for the definite integral.
The indefinite integral
and the general solution to the differential equation both describe the same thing: the collection of all functions with derivative f ( x ). Whether we're working in the context of integrals or of differential equations, we use the constant + Cto describe all antiderivatives or all solutions at once.
A definite integral is an integral of the form
The integral sign has limits of integration. A definite integral is a number. There's no need to write + C.
An indefinite integral is an integral of the form
.
There are no limits of integration on the integral sign. This indefinite integral is the family of all antiderivatives of f ( x ). Remember to write + C!
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In definite integrals we don't involve constant of integration because definite integral gives us a definite value at the point of calculation .In it we use two limits as a and b which is area under the curve .Definite integral has limits and numbers above and below the integral sign but in indefinite integrals which is unlimited or boundless we always add an unknown constant which is constant of integration because the derivative of F(x)+C is equal to the derivative of F(x).
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