# How do you use the limit process to find the area of the region between the graph #y=64-x^3# and the x-axis over the interval [1,4]?

# int_1^4 \ 64-x^3 \ dx = 513/4#

By definition of an integral, then

That is

And so:

Using the standard summation formula:

we have:

Using Calculus

If we use Calculus and our knowledge of integration to establish the answer, for comparison, we get:

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To find the area between the graph (y = 64 - x^3) and the x-axis over the interval ([1,4]) using the limit process, you would first divide the interval into subintervals of equal width. Then, you would choose sample points within each subinterval, typically either the left or right endpoint. Next, you would find the area of each rectangle formed by the subinterval width and the function value at the sample point. Finally, you would take the limit as the number of subintervals approaches infinity to get the exact area under the curve within the specified interval.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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