# Express the integral as a limit of Riemann sums?

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#\int_{4}^{12}[\ln(1+x^2)-\sin(x)]dx#

*Do not evaluate the limit.*

**Please don't use anything above Calculus I level.**

*Do not evaluate the limit.*

**Please don't use anything above Calculus I level.**

Here is a limit definition of the definite integral. (I'd guess it's the one you are using.)

Let's go one small step at a time.

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To express an integral as a limit of Riemann sums, you use the following formula:

[ \int_{a}^{b} f(x) , dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \cdot \Delta x ]

where:

- (a) and (b) are the lower and upper bounds of integration respectively,
- (f(x)) is the function being integrated,
- (n) is the number of partitions,
- (x_i^*) is a sample point in the (i)th subinterval,
- (\Delta x) is the width of each subinterval.

The Riemann sum is formed by summing the products of the function evaluated at the sample points (x_i^*) and the width of each subinterval (\Delta x), and as (n) approaches infinity, the Riemann sum approaches the integral.

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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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