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Find the value of the integral of f(x) = log10x from 5 to 11 by Simpson’s 1/3rd rule taking number of intervals, n = 6.
[Write the answer upto two decimal point]
- a)5.30
- b)5.35
- c)5.40
- d)5.45
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Find the value of the integral of f(x) = log10x from 5 to 11 by Simps...
To find the value of the integral of f(x) = log10x from 5 to 11 using Simpson's 1/3rd rule with 6 intervals, we need to follow the following steps:
Step 1: Calculate the interval width (h)
The interval width is given by the formula:
h = (b - a) / n
where a = 5 (lower limit), b = 11 (upper limit), and n = 6 (number of intervals).
Substituting the values, we have:
h = (11 - 5) / 6 = 6 / 6 = 1
Step 2: Calculate the values of f(x) at the given intervals
We need to calculate the values of f(x) = log10x at each interval. Since we have 6 intervals, we need to calculate the values at 7 points (including the lower and upper limits).
The values of x at each point are:
x0 = 5, x1 = 6, x2 = 7, x3 = 8, x4 = 9, x5 = 10, x6 = 11
Calculating the values of f(x) at each point:
f(x0) = log10(5) = 0.69897
f(x1) = log10(6) = 0.77815
f(x2) = log10(7) = 0.84510
f(x3) = log10(8) = 0.90309
f(x4) = log10(9) = 0.95424
f(x5) = log10(10) = 1.00000
f(x6) = log10(11) = 1.04139
Step 3: Calculate the integral using Simpson's 1/3rd rule
The integral can be calculated using the formula:
∫[a,b] f(x) dx ≈ (h/3) * [f(x0) + 4*f(x1) + 2*f(x2) + 4*f(x3) + 2*f(x4) + 4*f(x5) + f(x6)]
Substituting the values, we have:
∫[5,11] log10x dx ≈ (1/3) * [0.69897 + 4*0.77815 + 2*0.84510 + 4*0.90309 + 2*0.95424 + 4*1.00000 + 1.04139]
≈ (1/3) * [0.69897 + 3.11260 + 1.69020 + 3.61236 + 1.90848 + 4.00000 + 1.04139]
≈ (1/3) * 14.06310
≈ 4.68770
Rounding the answer to two decimal places, we get:
≈ 4.69
Therefore, the value of the integral of f(x) = log10x from 5 to 11 using Simpson's 1/3rd rule with 6 intervals is approximately 4.69.
Step 1: Calculate the interval width (h)
The interval width is given by the formula:
h = (b - a) / n
where a = 5 (lower limit), b = 11 (upper limit), and n = 6 (number of intervals).
Substituting the values, we have:
h = (11 - 5) / 6 = 6 / 6 = 1
Step 2: Calculate the values of f(x) at the given intervals
We need to calculate the values of f(x) = log10x at each interval. Since we have 6 intervals, we need to calculate the values at 7 points (including the lower and upper limits).
The values of x at each point are:
x0 = 5, x1 = 6, x2 = 7, x3 = 8, x4 = 9, x5 = 10, x6 = 11
Calculating the values of f(x) at each point:
f(x0) = log10(5) = 0.69897
f(x1) = log10(6) = 0.77815
f(x2) = log10(7) = 0.84510
f(x3) = log10(8) = 0.90309
f(x4) = log10(9) = 0.95424
f(x5) = log10(10) = 1.00000
f(x6) = log10(11) = 1.04139
Step 3: Calculate the integral using Simpson's 1/3rd rule
The integral can be calculated using the formula:
∫[a,b] f(x) dx ≈ (h/3) * [f(x0) + 4*f(x1) + 2*f(x2) + 4*f(x3) + 2*f(x4) + 4*f(x5) + f(x6)]
Substituting the values, we have:
∫[5,11] log10x dx ≈ (1/3) * [0.69897 + 4*0.77815 + 2*0.84510 + 4*0.90309 + 2*0.95424 + 4*1.00000 + 1.04139]
≈ (1/3) * [0.69897 + 3.11260 + 1.69020 + 3.61236 + 1.90848 + 4.00000 + 1.04139]
≈ (1/3) * 14.06310
≈ 4.68770
Rounding the answer to two decimal places, we get:
≈ 4.69
Therefore, the value of the integral of f(x) = log10x from 5 to 11 using Simpson's 1/3rd rule with 6 intervals is approximately 4.69.
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Community Answer
Find the value of the integral of f(x) = log10x from 5 to 11 by Simps...
We
∫abf(x)dx have,
h = (11−5)/6 = 1
We know that, by Simpson's 1/3 rd rule,
∫abf(x)dx
= h/3[s0+4s1+2s2]
Here, h = 1,a = 5,b = 11
s0 = f(x0) + f(x6) = 1.74
s1 = f(x1) + f(x3) + f(x5) = 2.681
s2 = f(x2) + f(x4) = 1.799
∫511log10xdx
= 1/3(1.74 + 10.724 + 3.598)
= 5.354
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Find the value of the integral of f(x) = log10x from 5 to 11 by Simpson’s 1/3rd rule taking number of intervals, n = 6.[Write the answer upto two decimal point]a)5.30b)5.35c)5.40d)5.45Correct answer is option 'B'. Can you explain this answer?
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