# How to you approximate the integral of # (t^3 +t) dx# from [0,2] by using the trapezoid rule with n=4?

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To approximate the integral of ( (t^3 + t) , dx ) from ( t = 0 ) to ( t = 2 ) using the trapezoid rule with ( n = 4 ), follow these steps:

- Divide the interval ( [0, 2] ) into ( n = 4 ) subintervals of equal width. In this case, each subinterval has a width of ( \Delta t = \frac{2 - 0}{4} = 0.5 ).
- Evaluate the function ( f(t) = t^3 + t ) at the endpoints of these subintervals and sum up the function values.
- Apply the trapezoid rule formula to each pair of adjacent function values and corresponding widths (which are all ( \Delta t = 0.5 )).
- Sum up the results from step 3 to get the approximation of the integral.

The trapezoid rule formula for a subinterval ( [a, b] ) is:

[ \text{Trapezoid Area} = \frac{b - a}{2} \times (f(a) + f(b)) ]

For each subinterval, the width is ( \Delta t = 0.5 ), and the formula becomes:

[ \text{Trapezoid Area} = 0.5 \times (f(a) + f(b)) ]

So, to approximate the integral using the trapezoid rule with ( n = 4 ):

- Evaluate ( f(t) = t^3 + t ) at ( t = 0, 0.5, 1.0, 1.5, ) and ( 2.0 ).
- Apply the trapezoid rule formula to each pair of adjacent function values and widths.
- Sum up the results obtained in step 2 to get the approximation of the integral.

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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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