How do I estimate the area under the graph of f(x) = #3 cos(x)# from #x=0# to #x=pi/2# using left and right endpoint methods?

Answer 1
You'll need a value for #n# to actually get an answer, but here's the method:
For your question #f(x)=3cos(x)#, #a=0#, and #b=pi/2#
Find #Delta x = (b-a)/n =(pi/2-0)/n=pi/(2n)#
Find all endpoints: start at #a# and successively add #Delta x# until you get to #b#:
#0, pi/(2n), (2pi)/(2n), (3pi)/(2n), * * * , ((n-1)pi)/(2n), (npi)/(2n)=pi/2#
Each rectangle has ares base times height, which will be #Delta x# times #f# at an enpoint

To use left endpoints , delete the last endpoint, above, because it is not a left endpoint

Rectangle 1 has area #Delta x *f(0) = (pi/(2n))3cos(0) #
Rectangle 2 has area #Delta x *f(pi/(2n)) = (pi/(2n))3cos(pi/(2n)) #
Rectangle 3 has area #Delta x *f((2pi)/(2n))=(pi/(2n))3cos(2pi/(2n))#
Rectangle 4 has area #Delta x *f((3pi)/(2n))=(pi/(2n))3cos(3pi/(2n))#

And so on up to

Rectangle n has area #Delta x *f(((n-1)pi)/(2n))=(pi/(2n))3cos(((n-1)pi)/(2n))#

Do the arithmetic and add the areas.

For right endpoints delete the case #x=0# and add the case #x= pi/2#
#(pi/(2n))3cos(pi/(2n))+pi/(2n))3cos(2pi/(2n)+(pi/(2n))3cos(3pi/(2n))+ * * * +(pi/(2n))3cos((pi)/2)#
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Answer 2

You can divide the range #0# to #pi/2# into a few rectangular strips and using left and right edges (endpoints) multiplied by the width of each strip calculate a minimum and a maximum area.

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Answer 3

To estimate the area under the graph of ( f(x) = 3 \cos(x) ) from ( x = 0 ) to ( x = \frac{\pi}{2} ) using the left and right endpoint methods, follow these steps:

  1. Divide the interval ([0, \frac{\pi}{2}]) into ( n ) subintervals of equal width. The width of each subinterval is given by ( \Delta x = \frac{\frac{\pi}{2} - 0}{n} = \frac{\pi}{2n} ).

  2. Choose ( n ) equally spaced points within the interval ([0, \frac{\pi}{2}]). For the left endpoint method, choose the left endpoint of each subinterval as the ( x )-coordinate. For the right endpoint method, choose the right endpoint of each subinterval.

  3. Evaluate the function ( f(x) = 3 \cos(x) ) at each chosen point.

  4. Compute the area of each rectangle formed by the height of the function at each chosen point and the width of the subinterval.

  5. Sum up the areas of all the rectangles to estimate the total area under the curve.

For the left endpoint method:

[ \text{Area} \approx \sum_{i=0}^{n-1} f(x_i) \Delta x ]

For the right endpoint method:

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

where ( x_i ) are the ( x )-coordinates of the chosen points within each subinterval.

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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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