# How do you use the limit process to find the area of the region between the graph #y=x^2+1# and the x-axis over the interval [0,3]?

# int_0^3 \ x^2+1 \ dx = 12 #

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 of the region between the graph ( y = x^2 + 1 ) and the x-axis over the interval ([0,3]) using the limit process, you can follow these steps:

- Divide the interval ([0,3]) into (n) equal subintervals of width ( \Delta x = \frac{3}{n}).
- Choose sample points (x_i^*) in each subinterval.
- Calculate the area of each rectangle formed by a subinterval and the function values at the sample points.
- Sum up the areas of all the rectangles to approximate the total area.
- Take the limit as (n) approaches infinity to find the exact area.

The formula for the area of each rectangle is ( A_i = f(x_i^*) \Delta x ), where ( f(x_i^*) ) represents the function value at the sample point (x_i^*) within the (i)th subinterval.

The total area is given by the limit of the sum of the areas of all rectangles as ( n ) approaches infinity:

[ \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x ]

In this case, with ( f(x) = x^2 + 1 ), you substitute the values of (x_i^*) into the function and sum up the areas of all rectangles. Then, take the limit as (n) approaches infinity to find the exact area.

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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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