2) Given the table below for selected values of f(x), use 6 trapezoids to estimate the value of (10)∫(1) f(x)dx
x 1 3 4 6 7 9 10
f(x) 4 8 6 10 10 12 16
To estimate the value of ∫(1) (10) f(x)dx using 6 trapezoids, we can use the Trapezoidal Rule. The Trapezoidal Rule is a numerical integration method that approximates the definite integral of a function using a series of trapezoids.
First, let's calculate the width of each trapezoid. Since we have 6 trapezoids and the interval is from 1 to 10, the width of each trapezoid is (10-1)/6 = 1.5.
Next, we need to calculate the areas of each trapezoid. The formula to calculate the area of a trapezoid is:
Area = (base1 + base2) / 2 * height
In this case, the bases of each trapezoid will be the values of f(x) at the consecutive points, and the height will be the width of the trapezoid.
Calculating the areas of each trapezoid:
Trapezoid 1:
base1 = f(1) = 4
base2 = f(1.5) = ?
height = 1.5 (width of each trapezoid)
Similarly, we can calculate the values of base2 for each trapezoid using the given table:
Trapezoid 1:
base1 = f(1) = 4
base2 = f(1.5) = 8
height = 1.5 (width of each trapezoid)
Trapezoid 2:
base1 = f(1.5) = 8
base2 = f(3) = 6
height = 1.5
Trapezoid 3:
base1 = f(3) = 6
base2 = f(4) = 10
height = 1.5
Trapezoid 4:
base1 = f(4) = 10
base2 = f(6) = 10
height = 1.5
Trapezoid 5:
base1 = f(6) = 10
base2 = f(7) = 12
height = 1.5
Trapezoid 6:
base1 = f(7) = 12
base2 = f(9) = 16
height = 1.5
Now that we have the values of the bases and the height for each trapezoid, we can calculate the areas:
Area 1 = (4 + 8) / 2 * 1.5 = 18
Area 2 = (8 + 6) / 2 * 1.5 = 15
Area 3 = (6 + 10) / 2 * 1.5 = 16.5
Area 4 = (10 + 10) / 2 * 1.5 = 15
Area 5 = (10 + 12) / 2 * 1.5 = 16.5
Area 6 = (12 + 16) / 2 * 1.5 = 24
Finally, to estimate the value of ∫(1) (10) f(x)dx using the Trapezoidal Rule, we sum up the areas of all the trapezoids:
Estimated value = Area 1 + Area 2 + Area 3 + Area 4 + Area 5 + Area 6
Estimated value = 18 + 15 + 16.5 + 15 + 16.5 + 24
Estimated value = 105
To estimate the value of the integral ∫(1)^(10) f(x) dx using 6 trapezoids, we can follow these steps:
Step 1: Find the width of each trapezoid.
The width of each trapezoid can be found by dividing the difference between the upper and lower limits of integration (10 - 1 = 9) by the number of trapezoids (6). So, the width of each trapezoid is 9/6 = 1.5.
Step 2: Calculate the area of each trapezoid.
To calculate the area of each trapezoid, we need to find the sum of the bases and multiply it by the height (which is the width of each trapezoid).
For the first trapezoid (from x = 1 to x = 2.5):
Base 1 = f(1) = 4
Base 2 = f(2.5) = (f(1) + f(3))/2 = (4 + 8)/2 = 6
Area of first trapezoid = (Base 1 + Base 2) * width = (4 + 6) * 1.5 = 15
For the second trapezoid (from x = 2.5 to x = 4):
Base 1 = f(2.5) = 6
Base 2 = f(4) = 6
Area of second trapezoid = (Base 1 + Base 2) * width = (6 + 6) * 1.5 = 18
For the third trapezoid (from x = 4 to x = 5.5):
Base 1 = f(4) = 6
Base 2 = f(5.5) = (f(4) + f(6))/2 = (6 + 10)/2 = 8
Area of third trapezoid = (Base 1 + Base 2) * width = (6 + 8) * 1.5 = 21
For the fourth trapezoid (from x = 5.5 to x = 7):
Base 1 = f(5.5) = 8
Base 2 = f(7) = 10
Area of fourth trapezoid = (Base 1 + Base 2) * width = (8 + 10) * 1.5 = 27
For the fifth trapezoid (from x = 7 to x = 8.5):
Base 1 = f(7) = 10
Base 2 = f(8.5) = (f(7) + f(9))/2 = (10 + 12)/2 = 11
Area of fifth trapezoid = (Base 1 + Base 2) * width = (10 + 11) * 1.5 = 31.5
For the sixth trapezoid (from x = 8.5 to x = 10):
Base 1 = f(8.5) = 11
Base 2 = f(10) = 16
Area of sixth trapezoid = (Base 1 + Base 2) * width = (11 + 16) * 1.5 = 40.5
Step 3: Sum up the areas of all the trapezoids.
Sum of all trapezoid areas = (Area of first trapezoid) + (Area of second trapezoid) + (Area of third trapezoid) + (Area of fourth trapezoid) + (Area of fifth trapezoid) + (Area of sixth trapezoid)
= 15 + 18 + 21 + 27 + 31.5 + 40.5
= 153
Therefore, using 6 trapezoids, the estimated value of the integral ∫(1)^(10) f(x) dx is 153.
you have the values. Just compute the area of the kth trapezoid using
ak = (f(xk)+f(xk+1))/2 * (xk+1-xk) for k=1..6
For example, a1 = (4+8)/2 * (3-1) = 12