How to solve this using limit of sum? #int_0^1xe^xdx#
Integrating by parts:
The left Riemann sum is then:
While the right Riemann sum is:
so that:
Now consider the geometric series:
Using the well known limit:
we can see that:
and then:
and we can conclude that:
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To solve the integral ( \int_{0}^{1} x e^{x} , dx ) using the limit of a sum (which is essentially the definition of the definite integral), you can approximate the integral using Riemann sums. We'll use right Riemann sums in this case.

Divide the interval ([0, 1]) into (n) subintervals of equal width ( \Delta x = \frac{1}{n} ).

Choose sample points ( x_i ) in each subinterval. We'll choose ( x_i = i \Delta x ) for the right Riemann sum.

The right Riemann sum is then given by: [ \sum_{i=1}^{n} f(x_i) \Delta x ] where ( f(x) = x e^x ).

Substitute ( x_i = i \Delta x ) and ( \Delta x = \frac{1}{n} ): [ \sum_{i=1}^{n} (i \Delta x) e^{i \Delta x} \frac{1}{n} ]

Simplify the expression inside the sum and the sum itself: [ \frac{1}{n} \sum_{i=1}^{n} i e^{i \Delta x} ]

Now, take the limit as ( n ) approaches infinity: [ \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} i e^{i \Delta x} ]
This limit represents the value of the integral ( \int_{0}^{1} x e^{x} , dx ). Calculating this limit exactly is challenging, so typically you would use numerical methods to approximate it.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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