How do you find the area using the trapezoidal approximation method, given #(x^2-x)dx#, on the interval [0,2] with n=4?

Answer 1

The interval #[0,2]# is split into #4# equal intervals of #1//2#. Pictured are the four functions, the intervals, and the areas created by making trapezoids from the function points.

The first three intervals are actually triangles since they have #0#-points that rest on the #x#-axis.

Using the points

  • #f(0)=0#
  • #f(1/2)=-1/4#
  • #f(1)=0#
  • #f(3/2)=3/4#
  • #f(2)=2#

we can calculate all the needed areas. The area for a trapezoid is given by #A=1/2h(b_1+b_2)#. Areas below the #x#-axis are negative.

#A_"total"=-1/2(1/2)(1/4)-1/2(1/2)(1/4)+1/2(1/2)(3/4)+1/2(1/2)(3/4+2)#

#=3/4#

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Answer 2

To find the area using the trapezoidal approximation method with ( n = 4 ) for the function ( x^2 - x ) on the interval ([0, 2]), you follow these steps:

  1. Divide the interval ([0, 2]) into ( n ) subintervals of equal width. Since ( n = 4 ), each subinterval will have width ( \Delta x = \frac{2 - 0}{4} = 0.5 ).

  2. Determine the function values at the endpoints of each subinterval. That is, evaluate ( f(0), f(0.5), f(1.0), f(1.5) ), and ( f(2.0) ) where ( f(x) = x^2 - x ).

  3. Use the trapezoidal rule formula to find the area of each trapezoid for each pair of adjacent points: [ A_i = \frac{h}{2}(f(x_i) + f(x_{i+1})) ] where ( h ) is the width of the subinterval and ( f(x_i) ) and ( f(x_{i+1}) ) are the function values at the endpoints of the subinterval.

  4. Sum up the areas of all the trapezoids to get the total approximate area under the curve within the interval ([0, 2]).

  5. Calculate: [ A_{\text{approx}} = A_1 + A_2 + A_3 + A_4 ]

  6. ( A_{\text{approx}} ) will give you the approximate area under the curve within the interval ([0, 2]) using the trapezoidal approximation method with ( n = 4 ).

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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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