Question

The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of

Answer 1

Solution

- The formula for finding the integral of a function using the trapezoidal rule is:

$A=\frac{\Delta x_1}{2}[f(x_0)+f(x_1)]+\frac{\Delta x_2}{2}[f(x_1)+f(x_2)]+...$

- Applying the formula, we have:

$\begin{gathered} \Delta x_1=1.1-1=0.1,\Delta x_2=1.2-1.1=0.1,\Delta x_3=1.5-1.2=0.3 \\ \Delta x_4=1.7-1.5=0.2,\Delta x_5=1.9-1.7=0.2,\Delta x_6=2.0-1.9=0.1 \\ \\ f(x_0)=f(1)=1 \\ f(x_1)=f(1.1)=2 \\ f(x_2)=f(1.2)=4 \\ f(x_3)=6 \\ f(x_4)=7 \\ f(x_5)=9 \\ f(x_6)=10 \end{gathered}$

- Thus, we can find the Integral as follows:

$\begin{gathered} A=\frac{0.1}{2}(2+1)+\frac{0.1}{2}(4+2)+\frac{0.3}{2}(6+4)+\frac{0.2}{2}(7+6)+\frac{0.2}{2}(9+7)+\frac{0.1}{2}(9+10) \\ \\ A=\frac{0.3}{2}+\frac{0.8}{2}+\frac{3}{2}+\frac{2.6}{2}+1.6+\frac{1.9}{2} \\ \\ A=5.9 \end{gathered}$

Final Answer

The integral is 5.9