How do you use the trapezoidal rule and five sub-intervals find approximation for this integral x=1 and x=3 for #1/x^2 dx#?

Answer 1
Approximate #int_1^3 1/x^2 dx# using trapezoids with #n=5#.
For this problems #a=1#, #b=3#, and #f(x)=1/x^2#
#Delta x = (b-a)/n = (3-1)/5 = 2/5 = 0.4# (I'll use fractions.)
#x_0 = a = 1#, to find, #x_1, x_2, . . . x_5# start at #x_0 = a = 1#,and add #Delta x# successively.
#x_0 = a = 1#, #x_2 = 7/5#, #x_3 = 9/5#, #x_4 = 11/5#, #x_5 = 13/5 = 3 = b#,

The approximation is:

#T = 1/2 Delta x (f(x_0) + 2f(x_1) +2f(x_2) +2f(x_3) +2f(x_4) +f(x_5))#

Plug in the values and do the arithmetic:

#T = 1/2(2/5)(1/(1)^2 + 2(1/(7/5)^2) +2(1/(9/5)^2)+2(1/(11/5)^2) +2(1/(13/5)^2)+(1/(3)^2))#
#= (1/5)(1+2(25/49)+2(25/81)+2(25/121) +2(25/169)#

Get a common denominator and simplify. (Yes, it really was this tedious before electronic calculators.)

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Answer 2

To use the trapezoidal rule with five sub-intervals to approximate the integral of ( \frac{1}{x^2} ) from ( x = 1 ) to ( x = 3 ), you first need to calculate the width of each sub-interval. Since you're dividing the interval ( [1, 3] ) into five equal parts, each sub-interval 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 sub-interval. For the trapezoidal rule, you'll average the function values at the endpoints of each sub-interval and multiply by the width of the sub-interval, summing these values for all sub-intervals.

Here's how to calculate the approximation:

  1. Evaluate ( f(x) ) at the endpoints and interior points of each sub-interval:

    • ( 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} )
  2. 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 sub-intervals.

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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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