The function f is continuous on the closed interval [0,6] and has values that are given in the table below.
x |0|2|4|6
f(x)|4|K|8|12
The trapezoidal approximation for(the integral):
6
S f(x) dx
1
found with 3 subintervals of equal length is 52. What is the value of K?
How do you do this with the formula
b-a/(2n) * f(x0) + 2(fx1) + f(xn)?
The formula for the area of a trapezoid is 1/2(b_1 + b_2)h where h = delta (x)
Therefore, A = 1/2(4+k)2 + 1/2(k+8)2 + 1/2(8+12)2 = 54
Solving that equation yields k = 10.
Ah, the trapezoidal approximation! Let's see if we can find the value of K by using the formula you provided.
First, let's break down the formula you mentioned:
(b - a) / (2n) * (f(x0) + 2 * ∑f(xi) + f(xn))
Here, 'b' represents the upper bound of the integral, 'a' is the lower bound, 'n' is the number of subintervals, 'x0' and 'xn' are the values at the lower and upper bounds, and '∑f(xi)' is the sum of all the values in between.
In this particular problem, we have b = 6, a = 0, n = 3, and the values of f(x) are given: f(0) = 4, f(2) = K, f(4) = 8, and f(6) = 12.
Now, we can plug these values into the formula and solve for K:
(6 - 0) / (2 * 3) * (4 + 2(K) + 8 + 12) = 52
Hmm, let me pull out my clown calculator for this one:
6/6 * (4 + 2K + 8 + 12) = 52
Simplifying:
24 + 12K + 48 = 312
12K + 72 = 312
12K = 240
K = 20
So, the value of K is 20! I hope that brought a smile to your face, or at least a chuckle.
To find the value of K, we can use the trapezoidal rule formula:
∫[a, b] f(x) dx ≈ (b - a) / (2n) * [f(x0) + 2(fx1) + ... + f(xn)],
where a and b are the limits of integration, n is the number of subintervals, and x0, x1, ..., xn are the points at which the function is evaluated.
In this specific case, we are given that:
∫[0, 6] f(x) dx ≈ 52,
n = 3 (3 subintervals),
a = 0, and
b = 6.
Let's substitute these values into the formula and solve for K step-by-step.
Step 1: Calculate the length of each subinterval.
The length of each subinterval is given by (b - a) / n.
In our case, (6 - 0) / 3 = 2.
Step 2: Identify the values of x0, x1, x2, and x3.
Since we have three subintervals, we will have four evaluation points: x0, x1, x2, and x3.
Given the table, we can see that x0 = 0, x1 = 2, x2 = 4, and x3 = 6.
Step 3: Substitute the values into the formula.
Using the trapezoidal rule formula, we have:
52 ≈ (6 - 0) / (2 * 3) * [f(0) + 2(f2) + f(4) + f(6)].
Substituting the known values, we get:
52 ≈ 6 / 6 * [4 + 2K + 8 + 12].
Step 4: Simplify and solve for K.
Expanding and simplifying further, we have:
52 ≈ [4 + 2K + 8 + 12].
Combining like terms, we get:
52 ≈ 24 + 2K.
Now, solve for K:
2K ≈ 52 - 24.
2K ≈ 28.
K ≈ 28 / 2.
K ≈ 14.
Therefore, the value of K is 14.
To find the value of K, we can use the trapezoidal approximation formula:
T = (b - a) / (2n) * [f(x0) + 2(fx1) + f(xn)]
Here, T is the trapezoidal approximation value, a is the lower bound of the interval (0 in this case), b is the upper bound of the interval (6 in this case), n is the number of subintervals (3 in this case), x0, x1, and xn are the x-values from the table (0, 2, and 6 respectively), and fx1 is the corresponding f-value for the x1 value.
Given that T = 52, a = 0, b = 6, and n = 3, we can substitute these values into the formula to solve for K.
52 = (6 - 0) / (2 * 3) * [4 + 2K + 12]
Now, simplify the equation:
52 = 6 / 6 * (16 + 2K)
52 = 16 + 2K
2K = 52 - 16
2K = 36
K = 36 / 2
K = 18
Therefore, the value of K in the given function is 18.