# How do you use the trapezoidal rule with n=6 to approximate the area between the curve #6sqrt(lnx)# from 1 to 4?

See the explanation section, below.

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To use the trapezoidal rule with (n = 6) to approximate the area between the curve (6\sqrt{\ln(x)}) from (1) to (4), follow these steps:

- Divide the interval ([1, 4]) into (n = 6) equal subintervals.
- Calculate the width of each subinterval, (\Delta x), using the formula: (\Delta x = \frac{b - a}{n}), where (a = 1) and (b = 4).
- Evaluate the function (6\sqrt{\ln(x)}) at each endpoint of the subintervals and at the midpoint of each subinterval.
- Apply the trapezoidal rule formula: (A \approx \frac{\Delta x}{2}\left[f(x_0) + 2\sum_{i=1}^{n-1}f(x_i) + f(x_n)\right]), where (x_0) is the lower limit of integration, (x_n) is the upper limit of integration, and (x_i) are the midpoints of the subintervals.
- Substitute the values of (f(x_i)) into the formula and calculate the sum to obtain the approximate area.

Using (n = 6), the subintervals are: ([1, 1.5]), ([1.5, 2]), ([2, 2.5]), ([2.5, 3]), ([3, 3.5]), and ([3.5, 4]).

The width of each subinterval is (\Delta x = \frac{4 - 1}{6} = \frac{1}{2}).

Evaluate the function at the endpoints and midpoints of the subintervals:

- (f(1) = 6\sqrt{\ln(1)} = 0)
- (f(1.5))
- (f(2))
- (f(2.5))
- (f(3))
- (f(3.5))
- (f(4))

Using these values, apply the trapezoidal rule formula to approximate the area.

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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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