How do you use the trapezoidal rule with n=10 to approximate the area between the curve #1/sqrt(1+x^3)# from 0 to 2?
Area is approximately
Divide the range
Calculate the value of
Calculate the area of each trapezoidal strip as
Sum the area of all the trapezoidal strips to get an approximation of the integral.
Theoretically this could be done by hand (if you need the arithmetic practice) but I chose to use a spreadsheet:
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To approximate the area under the curve (1/\sqrt{1+x^3}) from (x = 0) to (x = 2) using the trapezoidal rule with (n=10), follow these steps:

Divide the interval ([0, 2]) into (n=10) equal parts. This gives each subinterval a width of (\Delta x = \frac{20}{10} = 0.2).

Identify the endpoints of the subintervals: These are (x_0 = 0, x_1 = 0.2, x_2 = 0.4, \ldots, x_{10} = 2.0).

Evaluate the function at these points: Calculate (f(x_i) = 1/\sqrt{1+x_i^3}) for (i = 0, 1, \ldots, 10).

Apply the trapezoidal rule formula: The formula for the trapezoidal rule is
[ \text{Area} \approx \Delta x \left[\frac{f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n1}) + f(x_n)}{2}\right] ]
Substituting the values found:
[ \text{Area} \approx 0.2 \left[\frac{1/\sqrt{1+0^3} + 2\cdot1/\sqrt{1+(0.2)^3} + 2\cdot1/\sqrt{1+(0.4)^3} + \cdots + 2\cdot1/\sqrt{1+(1.8)^3} + 1/\sqrt{1+2^3}}{2}\right] ]

Calculate the sum and the area.
Let's compute the specific values for the function at these points:
 (f(x_0) = f(0) = 1/\sqrt{1+0^3} = 1)
 (f(x_1) = 1/\sqrt{1+(0.2)^3})
 (f(x_2) = 1/\sqrt{1+(0.4)^3})
 (\ldots)
 (f(x_{10}) = 1/\sqrt{1+2^3} = 1/\sqrt{9} = 1/3)
Now, compute each term (I will show the calculation for (f(x_1)) explicitly, and you would do similarly for the others):
 (f(x_1) = 1/\sqrt{1+(0.2)^3} \approx 1/\sqrt{1.008} \approx 0.996)
Perform these calculations for each (f(x_i)), then plug them into the trapezoidal rule formula and compute the sum to find the area approximation. This involves arithmetic that's best done with a calculator or software for precision.
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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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