How do you estimate the area under the graph of #f(x)= 2/x# on [1,5] into 4 equal subintervals and using right endpoints?

Answer 1

Use #f(x_1)*Deltax+ f(x_2)*Deltax+ f(x_3)*Deltax+ f(x_4)*Deltax#

Cut #[1,5}# into 4 subintervals of equal length #Delta x = (b-a)/n = (5-1)/4 = 1#
The endpoints of the subintervals are: #x_i = a+iDeltax# for #i = 0,1,2,3, . . . ,n#

In this question the subintervals are:

#[1,2]#, #[2,3]#, #{3, 4]#, [4,5]# and the RIGHT endpoints are:
#2, 3, 4, "and "5#

The heights at the endpoints are:

#f(2) = 2/2=1#
#f(3) = 2/3#
#f(4) = 2/4 = 1/2#
#f(5) = 2/5#

So the area is approximately:

#1*1+2/3*1+1/2*1+2/5*1 = 77/30#
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Answer 2

To estimate the area under the graph of ( f(x) = \frac{2}{x} ) on the interval ([1, 5]) using four equal subintervals and right endpoints, you can follow these steps:

  1. Divide the interval ([1, 5]) into four equal subintervals: ([1, 2]), ([2, 3]), ([3, 4]), and ([4, 5]).

  2. Calculate the width of each subinterval: ( \Delta x = \frac{b - a}{n} = \frac{5 - 1}{4} = \frac{4}{4} = 1 ).

  3. Determine the right endpoints of each subinterval: For the four subintervals, the right endpoints are ( x = 2 ), ( x = 3 ), ( x = 4 ), and ( x = 5 ).

  4. Evaluate the function ( f(x) = \frac{2}{x} ) at the right endpoints:

    • At ( x = 2 ): ( f(2) = \frac{2}{2} = 1 )
    • At ( x = 3 ): ( f(3) = \frac{2}{3} )
    • At ( x = 4 ): ( f(4) = \frac{2}{4} = \frac{1}{2} )
    • At ( x = 5 ): ( f(5) = \frac{2}{5} )
  5. Calculate the area of each rectangle formed by the function values and the width of the subinterval:

    • For the subinterval ([1, 2]), the area is ( 1 \times 1 = 1 )
    • For the subinterval ([2, 3]), the area is ( \frac{2}{3} \times 1 )
    • For the subinterval ([3, 4]), the area is ( \frac{1}{2} \times 1 = \frac{1}{2} )
    • For the subinterval ([4, 5]), the area is ( \frac{2}{5} \times 1 )
  6. Sum up the areas of all four rectangles to estimate the total area under the graph of ( f(x) ) on the interval ([1, 5]) using right endpoints.

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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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