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?

Answer 1

Area is approximately #2.09#

Divide the range #0# to #2# into #20# vertical strips at points #x_0:x_20# along the X-axis.

Calculate the value of #f(x_i) = 1/(sqrt(1+x_i^3)# for each point.

Calculate the area of each trapezoidal strip as
#A_i = (f(x_i)+f(x_(i+1)))/2 * width#
#color(white)("XXXXXXXX")# where #width# is the width of each strip (i.e. #2/20 = 0.1#)

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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Answer 2

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:

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

  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).

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

  4. 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_{n-1}) + 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] ]

  5. 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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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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