# How do you use the trapezoidal rule with n=4 to approximate the area between the curve #x ln(x+1)# from 0 to 2?

The trapezoidal rule states that the area under an integral can be approximated by the equation:

where:

So, we get:

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To use the trapezoidal rule with ( n = 4 ) to approximate the area between the curve ( x \ln(x + 1) ) from 0 to 2, follow these steps:

- Divide the interval [0, 2] into 4 equal subintervals. Since ( n = 4 ), each subinterval will have a width of ( \Delta x = \frac{2 - 0}{4} = 0.5 ).
- Calculate the function values at the endpoints and midpoint of each subinterval: ( x_0 = 0 ), ( x_1 = 0.5 ), ( x_2 = 1 ), ( x_3 = 1.5 ), ( x_4 = 2 ).
- Evaluate ( f(x) = x \ln(x + 1) ) at each of these points to get the corresponding function values: ( f(x_0) ), ( f(x_1) ), ( f(x_2) ), ( f(x_3) ), ( f(x_4) ).
- Use the trapezoidal rule formula to find the area under the curve within each subinterval: [ A_i = \frac{h}{2} \left( f(x_{i-1}) + f(x_i) \right) ] where ( h ) is the width of the subinterval and ( A_i ) is the area of the trapezoid for the ( i )-th subinterval.
- Sum up all the individual areas ( A_i ) to get the total approximate area under the curve.

Using the given formula and the calculated function values at each endpoint and midpoint of the subintervals, you can find the approximate area under the curve between 0 and 2 using the trapezoidal rule with ( n = 4 ).

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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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