# How do you evaluate the definite integral by the limit definition given #intsqrt(r^2-x^2)dx# from [-r,r]?

This can be integrated geometrically.

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To evaluate the definite integral ∫sqrt(r^2-x^2)dx from -r to r using the limit definition, follow these steps:

- Begin with the given integral: ∫sqrt(r^2-x^2)dx.
- To apply the limit definition, divide the interval [-r, r] into subintervals of equal width. Let Δx represent the width of each subinterval.
- Choose a representative point xi in each subinterval.
- Form the Riemann sum: R = Σsqrt(r^2-xi^2)Δx, where the sum is taken over all subintervals.
- Take the limit of the Riemann sum as the width of the subintervals approaches zero: lim(Δx→0) R.
- This limit will converge to the definite integral ∫sqrt(r^2-x^2)dx over the interval [-r, r].

By applying the limit definition of the definite integral, you'll arrive at the desired result without introducing additional complexities.

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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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