# Use Riemann sums to evaluate? : #int_0^(pi/2) \ sinx \ dx# ?

# int_0^(pi/2) \ sinx \ dx = 1 #

By definition of an integral, then

That is

And so:

If we were dealing with polynomials, we would now utilise standard summation formulas for powers, but as we are dealing with a sine summation those results will not help.

In order to evaluate the sine sum, We consider the following:

in combination with Euler's formula by taking

Then applying De Moivre's theorem:

and equating real and imaginary parts, we eventually find that:

And a standard calculus limit is:

Using Calculus

If we use Calculus and our knowledge of integration to establish the answer, for comparison, we get:

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To evaluate the integral ( \int_{0}^{\frac{\pi}{2}} \sin(x) , dx ) using Riemann sums, you can use the definition of the Riemann sum:

[ \sum_{i=1}^{n} f(x_i) \Delta x ]

where ( \Delta x ) is the width of each subinterval and ( x_i ) are the sample points within each subinterval. For this integral, you can choose ( n ) subintervals of equal width.

Choose ( n ) subintervals of equal width ( \Delta x = \frac{\pi}{2n} ). Then, the sample points for each subinterval will be ( x_i = \frac{\pi i}{2n} ) for ( i = 1, 2, ..., n ).

Now, evaluate ( f(x_i) = \sin(x_i) ) at each sample point and multiply by ( \Delta x ). Then, sum up all these products from ( i = 1 ) to ( n ).

[ \sum_{i=1}^{n} \sin\left(\frac{\pi i}{2n}\right) \frac{\pi}{2n} ]

As ( n ) approaches infinity, this Riemann sum approaches the value of the integral. So, to evaluate the integral, take the limit as ( n ) approaches infinity of this sum.

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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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