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?

Answer 1

#pi * f((0+pi)/2)+pi * f((pi+2pi)/2)=pi#, so the area under the curve is about #pi#. (In fact, it is exactly #pi#).

Since #n=pi#, we will be measuring rectangles with length #pi#. Since we're operating on the interval #[0,2pi]#, we will only have two rectangles.

The first rectangle is on the interval #[0,pi]#. The midpoint of #x=0# and #x=pi# is #x=(0+pi)/2=pi/2#.

At #x=pi/2# we have a rectangle with a base of #pi# and height of #f(pi/2)=cos(pi/2)+1/2=1/2#.

This rectangle has an area of #A=pi(1/2)=pi/2#.

The next rectangle will be on the interval #[pi,2pi]#, of which the midpoint is located at #x=(3pi)/2#.

The height of this rectangle is #f((3pi)/2)=cos((3pi)/2)+1/2=1/2#.

The area of the rectangle with base #pi# and height #1/2# is again #A=pi(1/2)=pi/2#.

The two rectangles we've found both have area #pi/2#, so the total area of both rectangles is #pi#, the approximate area under the curve.

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Answer 2

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:

  1. Determine the width of each rectangle: ( \Delta x = \frac{b - a}{n} = \frac{2\pi - 0}{\pi} = 2).

  2. 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 ).

  3. Evaluate ( f(x_i^*) ) for each midpoint to find the heights of the rectangles.

  4. Calculate the area of each rectangle: ( A_i = f(x_i^*) \Delta x ).

  5. Sum up the areas of all rectangles: ( \sum_{i=1}^n A_i ).

  6. 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.

  7. 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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Answer from HIX Tutor

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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