The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of 2∫1 f(x) dx .
x 1 1.1 1.3 1.6 1.7 1.8 2.0
f(x) 1 3 5 8 10 11 14
Just add up the values. I assume you know how to find the area of a trapezoid.
(f(1.0)+f(1.1))/2 * (1.1-1.0) +...+(f(1.8)+f(2.0))/2 * (2.0-1.8)
Too bad they didn't use equal widths. Then the sum would have been much easier:
1/6 (f(1) + 2*f(7/6) + ... + 2*f(11/6) + f(2))
slight oversight I think
(1/2)(1/6) [ f(1) + 2 f(7/6) etc
nice catch. I originally included the 1/2, but then I noticed that the widths were not all the same, and in my PS I left it out.
But I'm sure William picked right up on that... ... ...
To approximate the value of 2∫1 f(x) dx using a trapezoidal estimation, we need to divide the interval [1, 2] into smaller sub-intervals and calculate the area under the curve for each sub-interval.
Step 1: Determine the width of each sub-interval.
Since we are using 6 trapezoids, we divide the interval [1, 2] into 6 equal sub-intervals.
Width of each sub-interval = (Upper limit - Lower limit) / Number of trapezoids
= (2 - 1) / 6
= 1/6
Step 2: Calculate the area under the curve for each sub-interval.
To calculate the area under the curve for each sub-interval, we need to find the height of each trapezoid. The height of each trapezoid can be approximated by taking the average of the function values at the endpoints of each sub-interval.
Sub-interval 1: [1, 1 + 1/6]
Height of first trapezoid = (f(1) + f(1 + 1/6)) / 2
Sub-interval 2: [1 + 1/6, 1 + 2/6]
Height of second trapezoid = (f(1 + 1/6) + f(1 + 2/6)) / 2
Continue this process for all the sub-intervals.
Step 3: Calculate the sum of all the areas.
Sum up the calculated areas for each sub-interval to get the approximated value of the integral.
Approximated value of 2∫1 f(x) dx ≈ (Width of each sub-interval) * (Sum of all the areas)
Let's calculate the approximated value using the given function values:
Width of each sub-interval = 1/6
Height of first trapezoid = (f(1) + f(1 + 1/6)) / 2 = (1 + 3) /2 = 2
Area of first trapezoid = (1/6) * 2
Height of second trapezoid = (f(1 + 1/6) + f(1 + 2/6)) / 2 = (3 + 5) / 2 = 4
Area of second trapezoid = (1/6) * 4
Calculate the area and sum it up for each trapezoid using the same process.
Sum of all the areas = (1/6) * 2 + (1/6) * 4 + (1/6) * 4 + ... + (1/6) * (height of last trapezoid)
Finally, multiply the sum of all the areas by 2 to get the approximated value of the integral:
Approximated value of 2∫1 f(x) dx ≈ 2 * (Sum of all the areas)