# 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?

Draw 2 trapezoids under the curve, the first between

With

The first will have a width of

and an average height of

The second will also have a width of

and will have an average height of

The total area of the two trapezoids (approximating the area under the curve) is

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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:

- Divide the interval [1, 3] into ( n = 2 ) equal subintervals: [1, 2] and [2, 3].
- 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} ).

- 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.
- Substitute the values into the formula: [ A = \frac{2}{2} \left( 1 + 2\left(\frac{1}{4}\right) + \frac{1}{9} \right) ]
- 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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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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