Question

f(x) = = between x=4 and x = 7 using the "midpoint rule" with two rectangles of equal width.

Answer 1

EXPLANATION

Since we have the function f(x)= 1/x

$\int _a^bf\left(x\right)dx\:\approx \Delta \:x\left(f\left(\frac{x_0+x_1}{2}\right)+f\left(\frac{x_1+x_2}{2}\right)+f\left(\frac{x_2+x_3}{2}\right)+...+f\left(\frac{x_{n-1}+x_n}{2}\right)\right)$$\mathrm{where}\:\Delta \:x\:=\:\frac{b-a}{n}$

Given a=4, b=7, n=2

$Δx=\frac{7-4}{2}=\frac{3}{2}$

Divide the interval, 4<=x<=7 into 2 subintervals:

$x_0=4,\:x_1=\frac{11}{2},\:x_2=7$$=\frac{3}{2}\left(f\left(\frac{x_0+x_1}{2}\right)+f\left(\frac{x_1+x_2}{2}\right)\right)$

Calculate the subintervals:

$f\left(\frac{x_0+x_1}{2}\right)=f\left(\frac{4+\frac{11}{2}}{2}\right)$$=f\left(\frac{19}{4}\right)$$=\frac{1}{\frac{19}{4}}$$=\frac{4}{19}$$f\left(\frac{x_1+x_2}{2}\right)=f\left(\frac{\frac{11}{2}+7}{2}\right)$$=\frac{4}{25}$$=\frac{3}{2}\left(\frac{4}{19}+\frac{4}{25}\right)$

Applying the distributive property and adding fractions:

$=\frac{264}{475}$

In conclusion, the solution is 264/475