Would it be possible to use Reimann's sums to find the area enclosed by two curves ?
For example (don't solve this!), would we be able to use Reimann's sums to find the area between y=x^2 and let's say y=x+1?

The value of a definite integral is the limit of the Riemann sum over the integration interval. A Riemann sum can be used to approximate any definite integral to any desired degree of accuracy.

If the limit of the sum can be formulated, then the exact integral can be found.

Yes, a Riemann sum can be used to find the area between two curves.

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Additional comment

For this particular pair of curves, the limits of integration are irrational. This makes dividing the interval less convenient, but the general principles still hold. (The limits are (1±√5)/2.)

The attachment shows the Riemann sum of the difference between the curves evaluated at the midpoints of 20 evenly-spaced intervals. The error is about 0.13%. Using 200 intervals reduces the error to about 0.0015%.

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Would it be possible to use Reimann's sums to find the area enclosed by two curves ? For example (don't solve this!), would we be able to use Reimann's sums to find the area between y=x^2 and let's say y=x+1?

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Definite integral

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Would it be possible to use Reimann's sums to find the area enclosed by two curves ?
For example (don't solve this!), would we be able to use Reimann's sums to find the area between y=x^2 and let's say y=x+1?