Question:
gyazo.com/c8fe92564a8387ffc127233f6ad989a9
My work:
gyazo.com/27caca9994d03409268162130c61ce08
(I understand how to do trapezoidal approximation, but how would I go bout this while using only 4 sections?)
I realize I put 30 instead of 15. so the answer would actually be 2205!
When you solve using only 4 sections do you just ignore the other ones??
((f(0)+f(30))/2 * Δx) + ((f(30)+f(60))/2 * Δx) + ((f(60)+f(90))/2 * Δx) + ((f(90)+f(120))/2 * Δx)
(Δx)/(2) (f(0) + 2(f(30) + 2(f(60) + 2(f(90) + (f(120)
(30/2) (0+36 + 38 + 48 + 12)
A = 2010ft^2
You need to use 30, since you must cover the interval [0,120] with 4 subintervals. It just means that you use 2 width-15 intervals for each of your width-30 ones.
Your work looks good to me.
To solve this problem using trapezoidal approximation with only 4 sections, you need to divide the interval of integration into 4 equal-width subintervals.
Looking at the given work, it seems that you already have the interval [0, 1] divided into 5 sections. To adjust it to 4 sections, you can merge two adjacent sections into a single section.
Here's how you can do it:
1. Divide the interval [0, 1] into 5 equal-width subintervals.
2. Merge the second and third subintervals to create a single subinterval.
3. This merging will give you 4 equal-width subintervals in total.
Once you have the four sections, you can compute the trapezoidal approximation for each section separately, and then sum them up to get the total approximation:
1. Approximation for the first section:
- Calculate the area of the trapezoid formed by the function values at the endpoints of the section: f(0) and f(0.2).
- Divide it by 2 to get half the area of the trapezoid.
- Multiply this half area by the width of the section (0.2) to get the approximation for this section.
2. Approximation for the second section (which now combines the second and third subintervals):
- Calculate the area of the trapezoid formed by the function values at the endpoints: f(0.2) and f(0.4).
- Divide it by 2.
- Multiply this half area by the width of the section (0.2) to get the approximation for this section.
3. Approximation for the third section (which now corresponds to the original fourth subinterval):
- Calculate the area of the trapezoid formed by the function values at the endpoints: f(0.4) and f(0.6).
- Divide it by 2.
- Multiply this half area by the width of the section (0.2) to get the approximation for this section.
4. Approximation for the fourth section (which now corresponds to the original fifth subinterval):
- Calculate the area of the trapezoid formed by the function values at the endpoints: f(0.6) and f(1).
- Divide it by 2.
- Multiply this half area by the width of the section (0.2) to get the approximation for this section.
5. Sum up all the approximations from the four sections to get the total approximation:
Total Approximation = Approximation for Section 1 + Approximation for Section 2 + Approximation for Section 3 + Approximation for Section 4
By following these steps, you should be able to use trapezoidal approximation with only 4 sections to estimate the integral.