# How do you evaluate the definite integral #int x^2 dx# from #[0,1]#?

I am assuming that you do not yet have the Fundamental Theorem of Calculus available to evaluate this, but that you need to evaluate it from a definition.

I prefer to do this type of problem one small step at a time.

(We used summation formulas for the sums in the previous step.)

So,

There are a couple of ways to think about this limit :

OR

To finish the calculuation, we have

Using the Fundamental Theorem of Calculus

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To evaluate the definite integral ∫x^2 dx from [0,1], you use the fundamental theorem of calculus. The antiderivative of x^2 is (1/3)x^3. Evaluating this antiderivative at the upper and lower limits of integration (1 and 0, respectively) and subtracting yields the result.

∫x^2 dx from [0,1] = (1/3)x^3 evaluated from 0 to 1. Plugging in the upper limit: (1/3)(1)^3 = 1/3 Plugging in the lower limit: (1/3)(0)^3 = 0

Subtracting the lower limit value from the upper limit value: 1/3 - 0 = 1/3

Therefore, the value of the definite integral ∫x^2 dx from [0,1] is 1/3.

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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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