3 years ago

PLEASE HELP ME!!!!!!!!!

Mathematics

1 answer:

Genrish500 [490]3 years ago

7 0

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Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for the integr

prisoha [69]

Split up the interval [0, 2] into 4 subintervals, so that

$[0,2]=\left[0,\dfrac12\right]\cup\left[\dfrac12,1\right]\cup\left[1,\dfrac32\right]\cup\left[\dfrac32,2\right]$

Each subinterval has width $\dfrac{2-0}4=\dfrac12$ . The area of the trapezoid constructed on each subinterval is $\dfrac{f(x_i)+f(x_{i+1})}4$ , i.e. the average of the values of $x^2$ at both endpoints of the subinterval times 1/2 over each subinterval $[x_i,x_{i+1}]$ .

So,

$\displaystyle\int_0^2x^2\,\mathrm dx\approx\dfrac{0^2+\left(\frac12\right)^2}4+\dfrac{\left(\frac12\right)^2+1^2}4+\dfrac{1^2+\left(\frac32\right)^2}4+\dfrac{\left(\frac32\right)^2+2^2}4$