The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of integral from 1 to 2 f(x) dx
=.1*4/2+.1*7/2+.3*10/2+.2*13/2+.2(15/2)+.1*18/2
=4.40
5.75 is the right answer
sorry the table is
x 1 1.1 1.2 1.5 1.7 1.9 2.0
f(x) 1 3 4 6 7 8 10
I agree with your first line
=.1*4/2+.1*7/2+.3*10/2+.2*13/2+.2(15/2)+.1*18/2
= .1(2) + .1(3.5) + .3(5) + .2(6.5) + .2(7.5) + .1(9)
= .1(2+3.5+15+13+15+9)
= .1(57.5)
= 5.75
Well, it looks like someone has been having fun with numbers and formulas! But you know what they say, math can be quite the clown sometimes.
So, to estimate the value of the integral from 1 to 2 of f(x), you're using the trapezoidal method with 6 trapezoids. Now, let's put on our mathematical clown noses and give it a whirl!
First, let's break it down step by step:
The formula for the trapezoidal estimation is:
h * [f(a)/2 + f(x1) + f(x2) + ... + f(xn-1) + f(b)/2],
where h is the width of each trapezoid and n is the number of trapezoids.
Now, we can calculate it using the given values:
h = (2 - 1) / 6 = 1/6.
Since we have 6 trapezoids, we'll have n-1 = 5 intervals, which gives us:
x1 = 1 + 1/6 = 7/6
x2 = 1 + 2/6 = 4/3
x3 = 1 + 3/6 = 3/2
x4 = 1 + 4/6 = 5/3
x5 = 1 + 5/6 = 11/6
Now let's calculate:
integral ≈ 1/6 * [0.1/2 + 0.1*4/3 + 0.3*3/2 + 0.2*5/3 + 0.2*11/6 + 0.1*3],
And if we plug in those numbers, we get:
integral ≈ 1/6 * [0.05 + 0.133333 + 0.15 + 0.133333 + 0.183333 + 0.03],
Working it out, we get (drumroll, please):
integral ≈ 0.257222.
So, my clown friend, it looks like the estimated value of the integral from 1 to 2 of f(x) with 6 trapezoids is approximately 0.257222. Keep on clowning around with those numbers!
To approximate the value of the integral using the trapezoidal estimation, you need to follow these steps:
1. Start by dividing the interval [1, 2] into equally spaced subintervals. Since you are using 6 trapezoids, each subinterval will have a width of (2 - 1) / 6 = 1/6.
2. Evaluate the function f(x) at the endpoints of each subinterval. In this case, you have the following values:
f(1) = 4
f(1 + 1/6) = 7
f(1 + 2/6) = 10
f(1 + 3/6) = 13
f(1 + 4/6) = 15
f(1 + 5/6) = 18
f(2) = ?
Note: The value of f(2) is missing in the table. Please check for its value or assume a value before continuing the calculations.
3. Calculate the area of each trapezoid using the formula:
Area = (base1 + base2) * height / 2
In this case, the bases are the corresponding function values at the endpoints of each subinterval, and the height is the width of the subinterval.
4. Sum up the areas of all the trapezoids to get the approximation of the integral.
Let's compute the approximation using the given values:
Area = (.1 * (4 + 7) / 2) + (.1 * (7 + 10) / 2) + (.3 * (10 + 13) / 2) + (.2 * (13 + 15) / 2) + (.2 * (15 + 18) / 2) + (.1 * (18 + f(2)) / 2)
= (.1 * 11 / 2) + (.1 * 17 / 2) + (.3 * 23 / 2) + (.2 * 28 / 2) + (.2 * 33 / 2) + (.1 * (18 + f(2)) / 2)
= 5.5/2 + 8.5/2 + 34.5/2 + 14/2 + 33/2 + (18 + f(2))/20
= 2.75 + 4.25 + 17.25 + 7 + 16.5 + (18 + f(2))/2
= 48.75 + (18 + f(2))/2
= 48.75 + 9 + f(2)/2
= 57.75 + f(2)/2
Please provide the value of f(2) to calculate the final approximation of the integral from 1 to 2.