How do you use the trapezoidal rule with n=5 to approximate the area between the curve #y=(3x^2+4x+2)# from 0 to 3?

Question

Scarlett Allison · Accepted Answer

#49.296# square units

First, compute the length of your interval #=(3-0)/5=0.6# units Second, determine the #x#-values you need to use to compute #y# (i.e. #x=0.6, 1.2, 1.8, 2.4, 3.0#) Third, compute #y# using these #x#-values: #y=5.48,11.12,18.92,28.88,41.00# respectively Fourth, use the trapezoidal rule: #A=1/2 (0.6) [5.48 + 2(11.12) + 2(18.92) + 2(28.88) + 41.00]=49.296# square units

Ezekiel Alexander · Answer

undefined To use the Trapezoidal Rule with $ n = 5 $ to approximate the area between the curve $ y = 3x^2 + 4x + 2 $ from $ x = 0 $ to $ x = 3 $, follow these steps:

1. Divide the interval $ [0, 3] $ into $ n = 5 $ equal subintervals. The width of each subinterval ($ \Delta x $) will be $ \frac{3 - 0}{5} = 0.6 $.

2. Calculate the function values $ y_i $ at the endpoints and midpoint of each subinterval. For $ n = 5 $, you'll have 6 points: $ x_0 = 0, x_1 = 0.6, x_2 = 1.2, x_3 = 1.8, x_4 = 2.4, x_5 = 3.0 $.

3. Use the Trapezoidal Rule formula for each pair of adjacent points: 
$$ A_i = \frac{1}{2} \cdot \Delta x \cdot (y_{i} + y_{i+1}) $$
where $ A_i $ is the area of the trapezoid for the $ i^{th} $ subinterval.

4. Sum up the areas of all the trapezoids to get the total approximate area under the curve between $ x = 0 $ and $ x = 3 $.

5. The final result will be the approximation of the area between the curve and the x-axis within the specified interval using the Trapezoidal Rule with $ n = 5 $.