The table below gives selected values for the function f(x).
Use a trapezoidal estimation, with 6 trapezoids to approximate the value of .
x 1 1.1 1.2 1.5 1.7 1.9 2.0
f(x) 1 3 4 6 7 8 10
1*(.1+.1)*.5 etc
To estimate the value of ∫f(x)dx using a trapezoidal estimation with 6 trapezoids, we need to calculate the area of each trapezoid and sum them up.
The formula for the area of a trapezoid is A = (a + b) * h / 2, where a and b are the lengths of the parallel sides of the trapezoid and h is the height (the difference between the x-values).
Given the x-values and f(x) values in the table, we can calculate the area of each trapezoid as:
- Trapezoid 1:
- a = f(1) = 1
- b = f(1.1) = 3
- h = 1.1 - 1 = 0.1
- Area1 = (1 + 3) * 0.1 / 2 = 0.4
- Trapezoid 2:
- a = f(1.1) = 3
- b = f(1.2) = 4
- h = 1.2 - 1.1 = 0.1
- Area2 = (3 + 4) * 0.1 / 2 = 0.35
- Trapezoid 3:
- a = f(1.2) = 4
- b = f(1.5) = 6
- h = 1.5 - 1.2 = 0.3
- Area3 = (4 + 6) * 0.3 / 2 = 1.8
- Trapezoid 4:
- a = f(1.5) = 6
- b = f(1.7) = 7
- h = 1.7 - 1.5 = 0.2
- Area4 = (6 + 7) * 0.2 / 2 = 1.3
- Trapezoid 5:
- a = f(1.7) = 7
- b = f(1.9) = 8
- h = 1.9 - 1.7 = 0.2
- Area5 = (7 + 8) * 0.2 / 2 = 1.5
- Trapezoid 6:
- a = f(1.9) = 8
- b = f(2.0) = 10
- h = 2.0 - 1.9 = 0.1
- Area6 = (8 + 10) * 0.1 / 2 = 0.9
Finally, to estimate the value of ∫f(x)dx using trapezoidal estimation with 6 trapezoids, we sum up the areas of all the trapezoids:
Approximated value of ∫f(x)dx = Area1 + Area2 + Area3 + Area4 + Area5 + Area6
= 0.4 + 0.35 + 1.8 + 1.3 + 1.5 + 0.9
= 6.25
To approximate the value of the definite integral using the trapezoidal estimation method, you need to follow these steps:
1. Determine the width of each trapezoid. In this case, you have 6 trapezoids, so you divide the interval [1, 2] into 6 equal subintervals. The width of each subinterval is calculated by dividing the difference between the upper and lower limits by the number of subintervals: (2 - 1) / 6 = 1/6.
2. Calculate the area of each trapezoid. The area of a trapezoid is the average of the function values at the two endpoints, multiplied by the width of the interval. For example, the area of the first trapezoid is [(f(1) + f(1.1)) / 2] * (1/6).
3. Sum up the areas of all the trapezoids. Add the areas of each trapezoid together to get an approximation of the definite integral. For instance, the approximation of the definite integral would be the sum of the areas of all six trapezoids: [(f(1) + f(1.1)) / 2] * (1/6) + [(f(1.1) + f(1.2)) / 2] * (1/6) + [(f(1.2) + f(1.5)) / 2] * (1/6) + [(f(1.5) + f(1.7)) / 2] * (1/6) + [(f(1.7) + f(1.9)) / 2] * (1/6) + [(f(1.9) + f(2.0)) / 2] * (1/6).
4. Evaluate the expression. Substitute the given values of f(x) into the formula and perform the necessary calculations to find the numerical approximation of the definite integral.
By following these steps, you should be able to calculate the approximation of the definite integral using the trapezoidal estimation method with 6 trapezoids for the given function f(x).