Interpret the definite integral A = integral [a,b] dA = integral L(x)dx as computing the area of a region in the xy plane. Then one can think of the definite integral as:

A. "accumulating" all of the small segments of area "dA" from a to b
B. the antiderivative of L(x)
C. "accumulating" all of the small segments of area "dA" from a to b AND "accumulating" all of the small segments of area "L(x) dx" from a to b, where L(x) represents the length of a rectangle at a particular x value, and dx the width
D. the antiderivative of dA
E. "accumulating" all of the small segments of area "L(x) dx" from a to b, where L(x)r epresents the length of a rectangle at a particular x value, and dx the widt

It’s c

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it is indeed c. (I checked)

E. "accumulating" all of the small segments of area "L(x) dx" from a to b, where L(x) represents the length of a rectangle at a particular x value, and dx the width.

You could think of it as adding up all the little rectangles of width "dx" and height "L(x)" to find the total area. It's like stacking up the rectangles one by one to get the overall area from 'a' to 'b'. Just be careful not to trip over the rectangles, it can be quite the area-obstacle course!

The definite integral A = ∫[a,b] dA = ∫L(x)dx represents computing the area of a region in the xy plane. Let's go through the options and see which one best describes the interpretation of the definite integral in this context:

A. "accumulating" all of the small segments of area "dA" from a to b:
This option suggests that the definite integral accumulates the small segments of area "dA" from a to b. This interpretation aligns with the idea of adding up infinitely many small areas to find the total area of a region. So, option A is a possible interpretation.

B. the antiderivative of L(x):
The definite integral is related to the antiderivative through the Fundamental Theorem of Calculus. However, in the given context, we are specifically computing the area of a region, not finding the antiderivative. So, option B is not the most suitable interpretation.

C. "accumulating" all of the small segments of area "dA" from a to b AND "accumulating" all of the small segments of area "L(x) dx" from a to b, where L(x) represents the length of a rectangle at a particular x value, and dx the width:
This option combines both the accumulation of small segments of area "dA" and the accumulation of small segments of area "L(x) dx." The interpretation considers both the vertical and horizontal components involved in determining the area of rectangles. So, option C is a valid interpretation.

D. the antiderivative of dA:
Similar to option B, this option relates the definite integral to the concept of antiderivative. Again, since we are concerned with finding the area, this option is not the most appropriate interpretation.

E. "accumulating" all of the small segments of area "L(x) dx" from a to b, where L(x) represents the length of a rectangle at a particular x value, and dx the width:
This option solely focuses on the accumulation of small area segments "L(x) dx" from a to b. It considers the contribution of length and width in determining the area. Therefore, option E is also a reasonable interpretation.

To summarize, the interpretations that accurately describe the definite integral as computing the area of a region in the xy plane are:
A. "accumulating" all of the small segments of area "dA" from a to b
C. "accumulating" all of the small segments of area "dA" from a to b AND "accumulating" all of the small segments of area "L(x) dx" from a to b, where L(x) represents the length of a rectangle at a particular x value, and dx the width
E. "accumulating" all of the small segments of area "L(x) dx" from a to b, where L(x) represents the length of a rectangle at a particular x value, and dx the width