How do you use the trapezoidal rule with n=4 to approximate the area between the curve #y(t)=(t^3 +t)# from 0 to 2?
Refer to explanation
Hence we have n=4 we got 4 trapezoid hence we have that
From the trapezoidal rule we have that
So we have that
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To use the trapezoidal rule with ( n = 4 ) to approximate the area between the curve ( y(t) = t^3 + t ) from 0 to 2, follow these steps:

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

Calculate the function values at the endpoints of these subintervals. In this case, you will need to evaluate ( y(t) ) at ( t = 0, 0.5, 1, 1.5, ) and ( 2 ).

Use the trapezoidal rule formula to approximate the area under the curve within each subinterval:
[ \text{{Area of a single trapezoid}} = \frac{{\Delta t}}{2} \left[ y(t_0) + 2\sum_{i=1}^{n1} y(t_i) + y(t_n) \right] ]
where ( \Delta t ) is the width of each subinterval, ( y(t_0) ) and ( y(t_n) ) are the function values at the endpoints, and ( y(t_i) ) are the function values at the interior points of the subinterval.
 Sum up the areas of all the trapezoids to get the total approximate area under the curve.
In this case, you would compute the area of four trapezoids, each corresponding to one of the four subintervals within the interval [0, 2]. Then, sum up these areas to obtain the total approximation of the area between the curve ( y(t) = t^3 + t ) and the xaxis over the interval [0, 2].
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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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