How do you use the trapezoidal rule with n=6 to approximate the area between the curve #9 sqrt (ln x) # from 1 to 4?

Answer 1

See the explanation section below.

To approximate the Integral #int_a^b f(x) dx# using trapezoidal approximation with #n# intervals, use #T_n=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_(n-1))+f(x_n)] #
In this question we have: #f(x) = 9sqrt(lnx)# #{a,b] = [1, 4]#, and #n=6#.
So we get #Delta x = (b-a)/n = (4-1)/6 = 1/2 = 0.5#
The endpoints of the subintervals are found by beginning at #a=1# and successively adding #Delta x = 0.5# to find the points until we get to #x_n = b = 4#.
#x_0 = 1#, #x_1 = 1.5#, #x_2 = 2#, #x_3 = 2.5#, #x_4 = 3#, #x_5 = 3.5#, #x_6 = 4=b#,
Now apply the formula (do the arithmetic) for #f(x) = 9sqrt(lnx)#.
#T_6=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_5)+f(x_6)] #
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Answer 2

To use the trapezoidal rule with n = 6 to approximate the area between the curve 9√(ln x) from 1 to 4, follow these steps:

  1. Divide the interval [1, 4] into n = 6 equal subintervals. The width of each subinterval, Δx, is calculated as (b - a) / n, where a = 1 and b = 4.

  2. Compute the function values at the endpoints of each subinterval. This involves evaluating the function 9√(ln x) at x = 1, 4, 1.5, 2, 2.5, 3, and 3.5.

  3. Apply the trapezoidal rule formula for each pair of adjacent points:

    Area of each trapezoid = (width of subinterval) * (sum of function values at endpoints)

  4. Sum up the areas of all trapezoids to get the approximate total area under the curve between x = 1 and x = 4.

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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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