How do you use the Trapezoidal Rule and the Simpson's Rule when n=4 when approximating the integral # (5t + 6) dt# from [3,6]?

Question

How do you use the Trapezoidal Rule and the Simpson's Rule when n=4  when approximating the integral # (5t + 6) dt# from [3,6]?

Emily Ammerman · Accepted Answer

#int_3^6(5t+6)dt~~85.5#

#int_a^bf(t)dt~~(Deltax)/2(f(t_0)+2f(t_1)+2f(t_2)+...+2f(t_(n-1))+f(t_n))# Where #Deltat=(b-a)/2# We have #a=3,b=6,n=4# Therefore, #Deltat=(6-3)/4=3/4# Next, we divide [3,6] into four equal intervals with the length #3/4# This will give us the following endpoints: #3,15/4,9/2,21/4,6#

Now we simply evaluate the function at these endpoints; #f(t_0)=f(a)=f(3)=21# #2f(t_1)=2f(15/4)=99/2=49.5# #2f(t_2)=2f(9/2)=57# #2f(t_3)=2f(21/4)=129/2=64.5# #f(t_4)=f(b)=f(6)=36#

Now, we just sum up the values and multiply by #(Deltat)/2=3/8# #=3/8(21+49.5+57+64.5+36)=85.5#

Hope this helps :)

Cameron Alexander · Answer

undefined To approximate the integral $ \int_{3}^{6} (5t + 6) dt $ using the Trapezoidal Rule and Simpson's Rule with $ n = 4 $, we first need to divide the interval [3, 6] into subintervals. Since $ n = 4 $, there will be 4 subintervals of equal width.

1. **Subintervals**:
   Divide the interval [3, 6] into 4 subintervals:
   $$ \Delta t = \frac{{b - a}}{n} = \frac{{6 - 3}}{4} = 0.75 $$
   The subintervals are:
   $$ [3, 3.75], [3.75, 4.5], [4.5, 5.25], [5.25, 6] $$

2. **Calculate Function Values**:
   Evaluate the function $ f(t) = 5t + 6 $ at the endpoints and midpoints of each subinterval.

3. **Trapezoidal Rule**:
   Apply the Trapezoidal Rule formula for each subinterval:
   $$ \text{Trapezoidal Rule} = \frac{{\Delta t}}{2} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(t_i) + f(b) \right] $$
   Calculate the sum using the function values obtained in step 2.

4. **Simpson's Rule**:
   Apply Simpson's Rule formula for each pair of adjacent subintervals:
   $$ \text{Simpson's Rule} = \frac{{\Delta t}}{3} \left[ f(a) + 4 \sum_{i=1,3}^{n-1} f(t_i) + 2 \sum_{i=2,4}^{n-1} f(t_i) + f(b) \right] $$
   Calculate the sums using the function values obtained in step 2.

5. **Summation**:
   Add up the results obtained from applying Trapezoidal Rule and Simpson's Rule to get the approximations for the integral over the entire interval.

6. **Result**:
   The result will be the sum of the approximations obtained from Trapezoidal Rule and Simpson's Rule.