How do you Use the trapezoidal rule with #n=8# to approximate the integral #int_0^pix^2*sin(x)dx#?

Question

Christopher Agresta · Accepted Answer

The "trap rule" approximates the area by creating n trapezoids with their bases on the x-axis, top corners along the curve y = f(x), and then adding their areas together. Here we evaluate the function f(x) = x^2 sin(x) at 9 points along the interval from 0 to π, to make 8 intervals of  #Delta#x = π/8  for the trapezoid bases. x = 0, π/8, π/4, . . . , 7π/8, and π, then plug all these into the function f(x), to get the heights of the sides of the trapezoids: f(0) = 0^2 sin(0) = 0, f(π/8) = (π/8)^2 sin(π/8), etc. Now use the trapezoid area formula: #A_(trap) = b*(h_1+h_2)/(2)#
In our case #DeltaA = Deltax*(f(x_(i-1))+f(x_i))/(2)# 
and when you add all n of these together you get  #sum_(i=1)^n DeltaA = Deltax*{f(x_0)+2f(x_1)+…+2f(x_(n-1))+f(x_n)}#  because the middle terms appear twice, the right side of one trapezoid being the left side of the next. For our example, #A = sum_(i=1)^8= π/8[f(0)+2*f(π/8)+…+2*f((7π)/8)+f(π)]#. You get to evaluate each term to as much accuracy as you need to get your answer to the specified tolerance. Happy calculator plugging!  \another fine answer from dansmath/

Ezra Adams · Answer

undefined To use the trapezoidal rule with $ n = 8 $ to approximate the integral $ \int_0^{\pi} x^2 \sin(x) \, dx $, follow these steps:

1. Determine the interval width, $ h $, which is given by $ h = \frac{{b - a}}{n} $, where $ n $ is the number of subintervals (in this case, $ n = 8 $) and $ a $ and $ b $ are the lower and upper limits of integration (in this case, $ a = 0 $ and $ b = \pi $).

2. Calculate $ h $: $ h = \frac{{\pi - 0}}{8} = \frac{\pi}{8} $.

3. Set up the formula for the trapezoidal rule: 
$$ \text{Trapezoidal Rule} = h \left[ \frac{1}{2} f(x_0) + f(x_1) + f(x_2) + \ldots + f(x_{n-1}) + \frac{1}{2} f(x_n) \right] $$

4. Calculate the function values $ f(x_i) $ for each subinterval $ [x_i, x_{i+1}] $. In this case, $ f(x) = x^2 \sin(x) $.

5. Evaluate the sum using the trapezoidal rule formula.

6. Multiply the sum by $ h $ to obtain the approximation for the integral.

Following these steps will yield an approximation of the integral $ \int_0^{\pi} x^2 \sin(x) \, dx $ using the trapezoidal rule with $ n = 8 $.