How do you find the area using the trapezoidal approximation method, given #(x^2x)dx#, on the interval [0,2] with n=4?
The interval
The first three intervals are actually triangles since they have
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
#=3/4#
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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:

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 ).

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 ).

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.

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

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

( 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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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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