How do you estimate the area under the curve #f(x)=x^2-9# in the interval [-3, 3] with n = 6 using the trapezoidal rule?

Answer 1
With #n=6# over the interval #[-3,+3]# we have 6 trapezoids each with a width of #1# unit
The Sum of the Areas of these trapezoids is #sum_(x=-3)^(x=+2) (f(x)+f(x+1))/2xx(1)# or #(f(-3)+f(+3))/2 + sum_(x=-2)^(x=+2) f(x)#

#{: ( x, color(white)("xxxxxx"), f(x)=x^2-9), (-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) :}#

and our estimated area under the curve is #(-35)#
Note this area is negative because the entire curve between #[-3,+3]# is below the X-axis
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Answer 2

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:

  1. 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).

  2. 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).

  3. Use the trapezoidal rule formula to find the area of each trapezoid:

[ A_i = \frac{{f(x_{i-1}) + f(x_i)}}{2} \cdot \Delta x ]

for (i = 1, 2, ..., n).

  1. 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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Answer from HIX Tutor

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.

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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