How do you approximate of #int sinx(dx)# from #[0,pi]# by the trapezoidal approximation using n=10?

Question

Charles Arocha · Accepted Answer

See the explanation section.

To approximate the Integral #int_a^b f(x) dx# using trapezoidal approximation with #n# intervals. In this question we have: #f(x) = sinx# #{a,b] = [0, pi]#, and #n=10#. So we get #Delta x = (b-a)/n = (pi-0)/10 = pi/10# The endpoints of the subintervals are found by beginning at #a=0# and successively adding #Delta x = pi/10# to find the points until we get to #x_n = b = pi#. #x_0 = 0#, #x_1 = pi/10#, #x_2 = (2pi)/10#, #x_3 = (3pi)/10# . . . #x_9 = (9pi)/10#, and #x_10 = (10pi)/10 = 10 = b# Now apply the formula (do the arithmetic) for #f(x) = sinx#. #T_4=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_9)+f(x_10)] #

Christopher Agresta · Answer

undefined To approximate the integral of $ \sin(x) $ from $ 0 $ to $ \pi $ using the trapezoidal approximation with $ n = 10 $, follow these steps:

1. Divide the interval $[0,\pi]$ into $ n $ equal subintervals of width $ \Delta x = \frac{\pi}{n} $.
2. Use the trapezoidal rule formula to calculate the area of each trapezoid formed by adjacent subintervals and the function values at their endpoints.
3. Sum up the areas of all the trapezoids to approximate the total area.

The trapezoidal rule formula for each subinterval is:

$$ A_i = \frac{1}{2} \left( f(x_i) + f(x_{i+1}) \right) \Delta x $$

where $ f(x_i) $ and $ f(x_{i+1}) $ are the function values at the endpoints of the $ i $th subinterval.

In this case, with $ f(x) = \sin(x) $, you substitute the values of $ x_i $ and $ x_{i+1} $ into the function, calculate the area of each trapezoid, and then sum up the areas of all trapezoids to approximate the integral.