# How do you use limits to find the area between the curve #y=x^2# and the x axis from [0,5]?

Please see below.

Here is a limit definition of the area (actually for the definite integral). I will use what I think is frequently used notation in US textbooks.

Evaluate the sum

(We used a summation formula in the previous step.)

Rewrite before finding the limit

To finish the calculation, we have

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To find the area between the curve (y = x^2) and the x-axis from (x = 0) to (x = 5), you can integrate the absolute value of the function (y = x^2) over the interval ([0, 5]). This is because the function (y = x^2) is always non-negative over this interval, so the area will be the same as the integral of (|x^2|) over the interval ([0, 5]). Therefore, the area can be calculated as follows:

[A = \int_{0}^{5} |x^2| , dx]

[= \int_{0}^{5} x^2 , dx]

[= \left[\frac{x^3}{3}\right]_{0}^{5}]

[= \frac{5^3}{3} - \frac{0^3}{3}]

[= \frac{125}{3}]

So, the area between the curve (y = x^2) and the x-axis from (x = 0) to (x = 5) is ( \frac{125}{3} ) square units.

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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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