How do I estimate the area under the graph of #f(x)=25-x^2# from x=0 to x=5 using five rectangles and the right-endpoint method?

Answer 1

Estimate of area (using 5 right-endpoint rectangles = #70#

Using #5# rectangles (of width #1# each) the right end point will have a height of #color(white)("XXXX")##f(1) = 24# for the rectangle from #0# to #1# #color(white)("XXXX")##f(2) = 21# for the rectangle from #1# to #2# #color(white)("XXXX")##f(3) = 16# for the rectangle from #2# to #3# #color(white)("XXXX")##f(4) = 9# for the rectangle from #3# to #4# #color(white)("XXXX")##f(5) = 0# for the rectangle from #3# to #5#
and since each rectangle has a width of 1 the corresponding rectangles will have areas: #color(white)("XXXX")##24, 21, 16, 9, 0#
The sum of the areas of the #5# rectangles gives an estimate of the area under #f(x) = 25-x^2# from #0# to #5#
#24+21+16+9+0 = 70#
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Answer 2

To estimate the area under the graph of ( f(x) = 25 - x^2 ) from ( x = 0 ) to ( x = 5 ) using five rectangles and the right-endpoint method, follow these steps:

  1. Determine the width of each rectangle: ( \Delta x = \frac{{b - a}}{{n}} ), where ( a = 0 ) (starting point), ( b = 5 ) (ending point), and ( n = 5 ) (number of rectangles).

  2. Calculate the right endpoints for each rectangle: ( x_i = a + i \cdot \Delta x ) for ( i = 1, 2, 3, 4, 5 ).

  3. Evaluate the function at each right endpoint: ( f(x_i) ).

  4. Multiply each function value by the width of the rectangle: ( f(x_i) \cdot \Delta x ).

  5. Sum up the areas of all rectangles: ( A \approx \sum_{i=1}^{n} f(x_i) \cdot \Delta x ).

Plug in the values:

  1. Width of each rectangle: ( \Delta x = \frac{{5 - 0}}{{5}} = 1 ).

  2. Right endpoints: ( x_i = 0 + i \cdot 1 = i ) for ( i = 1, 2, 3, 4, 5 ).

  3. Evaluate function at right endpoints: ( f(x_i) = 25 - x_i^2 ).

  4. Multiply function value by width: ( f(x_i) \cdot \Delta x ).

  5. Sum up areas of rectangles: ( A \approx \sum_{i=1}^{5} f(x_i) \cdot \Delta x ).

Calculate each rectangle's area:

( A \approx (25 - (1)^2) \cdot 1 + (25 - (2)^2) \cdot 1 + (25 - (3)^2) \cdot 1 + (25 - (4)^2) \cdot 1 + (25 - (5)^2) \cdot 1 ).

( A \approx (25 - 1) + (25 - 4) + (25 - 9) + (25 - 16) + (25 - 25) ).

( A \approx 24 + 21 + 16 + 9 + 0 ).

( A \approx 70 ).

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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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