Question

Use the Trapezoidal Rule to approximate ∫43ln(x2+9) dx using n=3. Round your answer to the nearest hundredth.

Answer 1

The Trapezoidal rule formula is given to be:

`\begin{gathered} \int_a^bf(x)dx\approx(\triangle x)/(2)(f(x_o)+2f(x_1)+2f(x_2)+2f(x_3)+...+2f(x_(n-1))+f(x_n) \\ where \\ \triangle x=(b-a)/(n) \end{gathered}`

The question gives:

`\begin{gathered} f(x)=\ln(x^2+9) \\ a=3 \\ b=4 \\ n=3 \\ \therefore \\ \triangle x=(1)/(3) \end{gathered}`

Therefore, divide the interval into n = 3 subintervals of length 1/3 with the following endpoints:

`a=3,(10)/(3),(11)/(3),4`

Evaluate the function at the endpoints:

`\begin{gathered} f(x_0)=f(3)=2.89 \\ 2f(x_1)=2f((10)/(3))=6.00 \\ 2f(x_2)=2f((11)/(3))=6.22 \\ f(x_3)=f(4)=3.22 \end{gathered}`

Sum up the calculated values and multiply by Δx/2:

`\Rightarrow(1)/(3*2)(2.89+6.00+6.22+3.22)=3.06`

Therefore, the answer will be:

`\int_3^4\ln(x^2+9)dx\approx3.06`