How do you use the trapezoidal rule with n=2 to approximate the area under the curve #y=1/x^2# from 1 to 3?

Answer 1

Draw 2 trapezoids under the curve, the first between #x=1# and #x=2# and the second between #x=2# and #x=3#; use the sum of the areas to approximate the required area.

With #n=2# we have 2 trapezoids
The first will have a width of #1# (the distance on the X-axis between 1 and 2)
and an average height of #(1/1^2 + 1/2^2)/2 = 5/8#

The second will also have a width of #1# (the distance between 2 and 3)
and will have an average height of #(1/2^2+1/3^2)/2 = 13/36#

The total area of the two trapezoids (approximating the area under the curve) is
#color(white)("XXXX")##5/8 + 13/36 = (45+26)/72 = 71/72#

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Answer 2

To use the trapezoidal rule with ( n = 2 ) to approximate the area under the curve ( y = \frac{1}{x^2} ) from ( x = 1 ) to ( x = 3 ), follow these steps:

  1. Divide the interval [1, 3] into ( n = 2 ) equal subintervals: [1, 2] and [2, 3].
  2. Compute the function values at the endpoints of these subintervals:
    • At ( x = 1 ), ( y = \frac{1}{1^2} = 1 ).
    • At ( x = 2 ), ( y = \frac{1}{2^2} = \frac{1}{4} ).
    • At ( x = 3 ), ( y = \frac{1}{3^2} = \frac{1}{9} ).
  3. Calculate the area of each trapezoid formed by adjacent points using the formula: [ A = \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) ] where ( h ) is the width of each subinterval, ( f(x_0) ) and ( f(x_n) ) are the function values at the endpoints, and ( f(x_i) ) are the function values at the interior points.
  4. Substitute the values into the formula: [ A = \frac{2}{2} \left( 1 + 2\left(\frac{1}{4}\right) + \frac{1}{9} \right) ]
  5. Perform the calculations to find the approximate area under the curve.

[ A \approx \frac{1}{2} \left( 1 + \frac{1}{2} + \frac{1}{9} \right) = \frac{1}{2} \left( \frac{49}{36} \right) = \frac{49}{72} ]

So, the approximate area under the curve using the trapezoidal rule with ( n = 2 ) is ( \frac{49}{72} ).

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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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