How do you find the area using the trapezoidal approximation method, given #cos(x^2)#, on the interval [0, 1] with n=8?
and so we can conclude that
The values of
Using the trapezoidal rule:
# int_a^bydx ~~ h/2{(y_0+y_n)+2(y_1+y_2+...+y_(n1))}#
We have:
# int_(0)^(1) cos(x^2)dx ~~ 0.125/2 { 1 + 0.5403#
# " " + 2(0.99988 + 0.99805 + 0.99013 #
# " " + 0.96891 + 0.92467 + 0.84592 #
# " " + 0.72095)}#
# " " = 0.0625 { 1.5403 + 2( 6.44851 )}#
# " " = 0.0625 { 1.5403 + 12.89702 }#
# " " = 0.0625 { 14.43733 }#
# " " = 0.90233#
and so we can conclude that
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To find the area using the trapezoidal approximation method, given ( \cos(x^2) ) on the interval ([0, 1]) with ( n = 8 ), follow these steps:

Divide the interval ([0, 1]) into ( n ) subintervals of equal width. Since ( n = 8 ), each subinterval will have a width of ( \Delta x = \frac{10}{8} = \frac{1}{8} ).

Calculate the function values ( \cos(x^2) ) at the endpoints of each subinterval.

Apply the trapezoidal rule formula for each subinterval:
[ \text{Area of a single trapezoid} = \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n1}) + f(x_n) \right) ]
where ( \Delta x ) is the width of each subinterval, ( x_0 ) and ( x_n ) are the endpoints of the interval, and ( f(x_i) ) are the function values at the endpoints.
 Sum up the areas of all trapezoids to find the total approximate area under the curve.
Using this method, you can approximate the area under the curve ( \cos(x^2) ) on the interval ([0, 1]) with ( n = 8 ).
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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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