The function f is continuous on the closed interval [1,7] and has values in the table below
x f(x)
1 10
4 30
6 40
7 20
Usinf the subintervals [1,4] [4,6] [6,7] what is the trapezoidal apporx. of the integral from 1 to 7 of f(x)dx?
The answer: 160
I tried doing each interval as separate integrals and then using TRAP and then adding them but I got 180. How do I get the right answer?
The trapezoidal approximation of the integral from 1 to 7 of f(x)dx is calculated by adding the area of the three trapezoids formed by the three subintervals. The area of each trapezoid is calculated by taking the average of the two endpoints of the interval and multiplying it by the length of the interval.
For the interval [1,4], the area of the trapezoid is (10 + 30)/2 * (4 - 1) = 80.
For the interval [4,6], the area of the trapezoid is (30 + 40)/2 * (6 - 4) = 40.
For the interval [6,7], the area of the trapezoid is (40 + 20)/2 * (7 - 6) = 40.
Adding the areas of the three trapezoids gives a total of 160, which is the trapezoidal approximation of the integral from 1 to 7 of f(x)dx.
Well, it seems like you're having a bit of trouble with your calculations. Don't worry, I'm here to help you out with a dash of humor!
To find the trapezoidal approximation of the integral, you need to calculate the area of trapezoids formed by each subinterval. Let's break it down step by step:
1. For the first subinterval [1,4]:
- The bases of the trapezoid are 10 and 30 (the values of f(x) at x=1 and x=4, respectively).
- The height of the trapezoid is 3 (the difference between 4 and 1).
- So, the area of this trapezoid is ((10 + 30) / 2) * 3 = 40.
2. For the second subinterval [4,6]:
- The bases of the trapezoid are 30 and 40.
- The height of the trapezoid is 2.
- So, the area of this trapezoid is ((30 + 40) / 2) * 2 = 70.
3. For the third subinterval [6,7]:
- The bases of the trapezoid are 40 and 20.
- The height of the trapezoid is 1.
- So, the area of this trapezoid is ((40 + 20) / 2) * 1 = 30.
Now, all you need to do is add up the areas of the three trapezoids: 40 + 70 + 30 = 140.
Oh dear, it seems like I made a calculation error. My clown nose is blushing! It should be 40 + 70 + 30 = 140, not 160. So, the correct trapezoidal approximation of the integral from 1 to 7 of f(x)dx is 140.
Keep up the good work and remember, even when the calculations make you feel like a clown, it's all part of the learning circus!
To find the trapezoidal approximation of the integral from 1 to 7 of f(x)dx using the given subintervals [1,4], [4,6], and [6,7], we can follow these steps:
Step 1: Determine the length of each interval.
The length of the interval [1,4] is 4 - 1 = 3.
The length of the interval [4,6] is 6 - 4 = 2.
The length of the interval [6,7] is 7 - 6 = 1.
Step 2: Calculate the average of the function values at the endpoints in each interval.
For the interval [1,4], the average of the function values is (10 + 30) / 2 = 20.
For the interval [4,6], the average of the function values is (30 + 40) / 2 = 35.
For the interval [6,7], the average of the function values is (40 + 20) / 2 = 30.
Step 3: Calculate the area of each trapezoid.
For the interval [1,4], the area of the trapezoid is (3 * (20 + 10)) / 2 = 45.
For the interval [4,6], the area of the trapezoid is (2 * (35 + 30)) / 2 = 65.
For the interval [6,7], the area of the trapezoid is (1 * (30 + 20)) / 2 = 25.
Step 4: Add up the areas of all the trapezoids to get the total approximate area.
Total approximate area = 45 + 65 + 25 = 135.
Therefore, the trapezoidal approximation of the integral from 1 to 7 of f(x)dx using the given subintervals is 135, not 160 as you mentioned. Check your calculations again to see where the mistake might have occurred.
To find the trapezoidal approximation of the integral from 1 to 7 of f(x) dx using the given subintervals [1,4], [4,6], and [6,7], you need to calculate the area of trapezoids for each subinterval and then sum them up.
Let's go through the process step by step:
1. Start by calculating the base of each trapezoid, which is equal to the length of each subinterval.
- For [1,4], the base is 4 - 1 = 3.
- For [4,6], the base is 6 - 4 = 2.
- For [6,7], the base is 7 - 6 = 1.
2. Calculate the heights of each trapezoid using the function values provided in the table.
- For [1,4], the heights are 10 and 30 corresponding to the function values at x = 1 and x = 4, respectively.
- For [4,6], the heights are 30 and 40 corresponding to the function values at x = 4 and x = 6, respectively.
- For [6,7], the heights are 40 and 20 corresponding to the function values at x = 6 and x = 7, respectively.
3. Calculate the areas of each trapezoid using the formula: (base / 2) * (height1 + height2).
- For [1,4], the area is (3 / 2) * (10 + 30) = 3 * 20 = 60.
- For [4,6], the area is (2 / 2) * (30 + 40) = 1 * 70 = 70.
- For [6,7], the area is (1 / 2) * (40 + 20) = 0.5 * 60 = 30.
4. Finally, sum up the areas of all the trapezoids.
- 60 + 70 + 30 = 160.
Therefore, the trapezoidal approximation of the integral from 1 to 7 of f(x) dx using the given subintervals is 160, not 180.