How do I estimate the area under the graph of #f(x)=25x^2# from x=0 to x=5 using five rectangles and the rightendpoint method?
Estimate of area (using 5 rightendpoint rectangles =
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To estimate the area under the graph of ( f(x) = 25  x^2 ) from ( x = 0 ) to ( x = 5 ) using five rectangles and the rightendpoint method, follow these steps:

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).

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

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

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

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

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

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

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

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

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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When evaluating a onesided 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 onesided 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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