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If f has a discontinuity at c, where a < c < b, and both integral a through c f(x)dx and integral c through b f(x)dx are convergent, then we define integral a through b f(x) dx = ?

A) Convergent
B) Divergent
C) Undefined
D) Zero

User Taptronic
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1 Answer

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Final answer:

The integral of a function with a discontinuity at point c can be defined as the sum of the integrals from a to c and from c to b, given that both are convergent. Hence, the integral from a to b of f(x) is convergent, which corresponds to option A) Convergent.

Step-by-step explanation:

When dealing with integrals and discontinuities in a function f, it is possible to approach integration piecewise, especially when the discontinuity occurs at a point c where a < c < b. We are given that both the integral from a to c, ∫ac f(x) dx, and the integral from c to b, ∫cb f(x) dx, are convergent. This implies that each part exists as a finite number. According to the rules of integration, we can add up these two convergent integrals to define the integral from a to b.

The sum of the areas represented by the integrals on each side of the discontinuity is therefore just the sum of the two integrals. As a result, ∫ab f(x) dx is defined as the sum of the integrals on either side of c. So if we defined ∫ac f(x) dx as P, and ∫cb f(x) dx as Q, then ∫ab f(x) dx would be P + Q. This makes the answer A) Convergent. The limit point of the discontinuity does not contribute to the area since it has no width, as per the idea that P(x = c) = 0 because an individual point does not have an area.

User Mora
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