How do you estimate the area under the curve #f(x)=x^29# in the interval [3, 3] with n = 6 using the trapezoidal rule?
#{: ( x, color(white)("xxxxxx"), f(x)=x^29), (3, color(white)("xxxxxx"), 0), (2, color(white)("xxxxxx"), 5), (1, color(white)("xxxxxx"), 8), ( +0, color(white)("xxxxxx"), 9), (+1, color(white)("xxxxxx"), 8), (+2, color(white)("xxxxxx"), 5), (+3 color(white)("xxxxxx"),, 0) :}#
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To estimate the area under the curve ( f(x) = x^2  9 ) in the interval ([3, 3]) with (n = 6) using the trapezoidal rule, follow these steps:

Divide the interval ([3, 3]) into (n) subintervals of equal width. Since (n = 6), each subinterval will have a width of (\Delta x = \frac{{3  (3)}}{6} = 1).

Calculate the function values at the endpoints of each subinterval. This will give you (f(x_i)) for (i = 0, 1, 2, ..., n), where (x_i = 3 + i \cdot \Delta x) for (i = 0, 1, 2, ..., n).

Use the trapezoidal rule formula to find the area of each trapezoid:
[ A_i = \frac{{f(x_{i1}) + f(x_i)}}{2} \cdot \Delta x ]
for (i = 1, 2, ..., n).
 Sum up the areas of all the trapezoids to get the estimated area under the curve:
[ A \approx \sum_{i=1}^{n} A_i ]
Substituting the values obtained in steps 2 and 3 into this formula will give you the estimated area under the curve using the trapezoidal rule with (n = 6).
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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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