# How do you use limits to evaluate #int x^2dx# from [0,4]?

Please see below.

Here is a limit definition of the definite integral. (I hope it's the one you are using.) I will use what I think is somewhat standard 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 evaluate the integral ∫x^2 dx from 0 to 4 using limits, you can follow these steps:

- Start with the indefinite integral: ∫x^2 dx.
- Find the antiderivative of x^2, which is (1/3)x^3.
- Evaluate the antiderivative at the upper limit of integration (4) and subtract the value when evaluated at the lower limit of integration (0).
- Substitute the upper limit into the antiderivative: (1/3)(4)^3.
- Substitute the lower limit into the antiderivative: (1/3)(0)^3.
- Subtract the result of evaluating the antiderivative at the lower limit from the result of evaluating it at the upper limit.

So, the integral of x^2 from 0 to 4 equals:

(1/3)(4)^3 - (1/3)(0)^3 = (1/3)(64) - (1/3)(0) = (64/3) - 0 = 64/3.

Therefore, the value of the integral ∫x^2 dx from 0 to 4 is 64/3.

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