How do you use Riemann sums to evaluate the area under the curve of #f(x)=cosx+0.5# on the closed interval [0,2pi], with n=pi rectangles using midpoints?
Since
The first rectangle is on the interval
At
This rectangle has an area of
The next rectangle will be on the interval
The height of this rectangle is
The area of the rectangle with base
The two rectangles we've found both have area
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To use Riemann sums to evaluate the area under the curve of (f(x) = \cos(x) + 0.5) on the closed interval ([0, 2\pi]) with (n = \pi) rectangles using midpoints, follow these steps:
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Determine the width of each rectangle: ( \Delta x = \frac{b - a}{n} = \frac{2\pi - 0}{\pi} = 2).
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Determine the midpoints of each rectangle: [ x_i^* = a + \frac{(i - 0.5) \Delta x}{n} = \frac{(i - 0.5)2}{\pi} ] where ( i = 1, 2, \ldots, n ).
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Evaluate ( f(x_i^*) ) for each midpoint to find the heights of the rectangles.
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Calculate the area of each rectangle: ( A_i = f(x_i^*) \Delta x ).
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Sum up the areas of all rectangles: ( \sum_{i=1}^n A_i ).
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This sum represents an approximation of the area under the curve. As ( n ) approaches infinity, the approximation becomes more accurate, approaching the actual area under the curve.
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Calculate the final value by taking the limit as ( n ) approaches infinity: [ \lim_{n \to \infty} \sum_{i=1}^n A_i ]
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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.
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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