How do you use the trapezoidal rule and five subintervals find approximation for this integral x=1 and x=3 for #1/x^2 dx#?
The approximation is:
Plug in the values and do the arithmetic:
Get a common denominator and simplify. (Yes, it really was this tedious before electronic calculators.)
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To use the trapezoidal rule with five subintervals to approximate the integral of ( \frac{1}{x^2} ) from ( x = 1 ) to ( x = 3 ), you first need to calculate the width of each subinterval. Since you're dividing the interval ( [1, 3] ) into five equal parts, each subinterval has a width of ( \Delta x = \frac{3  1}{5} = 0.4 ).
Next, evaluate the function ( f(x) = \frac{1}{x^2} ) at the endpoints and interior points of each subinterval. For the trapezoidal rule, you'll average the function values at the endpoints of each subinterval and multiply by the width of the subinterval, summing these values for all subintervals.
Here's how to calculate the approximation:

Evaluate ( f(x) ) at the endpoints and interior points of each subinterval:
 ( f(1) = 1^2 = 1 )
 ( f(1.4) = \frac{1}{(1.4)^2} )
 ( f(1.8) = \frac{1}{(1.8)^2} )
 ( f(2.2) = \frac{1}{(2.2)^2} )
 ( f(2.6) = \frac{1}{(2.6)^2} )
 ( f(3) = \frac{1}{3^2} = \frac{1}{9} )

Use the trapezoidal rule formula: [ \text{Approximation} = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + f(x_5)] ]
Substitute the values: [ \text{Approximation} = \frac{0.4}{2} [1 + 2f(1.4) + 2f(1.8) + 2f(2.2) + 2f(2.6) + \frac{1}{9}] ]
Evaluate ( f(x) ) at ( x = 1.4, 1.8, 2.2, ) and ( 2.6 ), then calculate the approximation using the formula.
This calculation will provide an approximation of the integral of ( \frac{1}{x^2} ) from ( x = 1 ) to ( x = 3 ) using the trapezoidal rule with five subintervals.
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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.
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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