Question

Use the Trapezoidal Rule to approximate ∫73x2+6‾‾‾‾‾‾√dx using n=3. Round your answer to the nearest hundredth.

Answer 1

Answer: 22.39

Given:

`\int_3^7√(x^2+6)dx\text{ }n=3`

The trapezoidal rule states that:

`\int_a^bf(x)dx\approx(\Delta x)/(2)(f(x_0)+2f(x_1)+2f(x_2)+2f(x3)+...+f(x_n)`

Where:

`\Delta x=(b-a)/(n)`

From the given, we know that:

`\begin{gathered} f(x)=√(x^2+6) \\ a=3 \\ b=7 \\ n=3 \end{gathered}`

With this, we know that:

`\Delta x=(7-3)/(3)=(4)/(3)`

We will then divide the interval [3,7] into n=3 subintervals of 4/3, which will give us:

`3,(13)/(3),(17)/(3),7`

Now, we will evaluate the function at these endpoints:

We will then sum up the values and multiply by Δx/2:

`\begin{gathered} (\Delta x)/(2)=((4)/(3))/(2)=(2)/(3) \\ \Rightarrow(2)/(3)(√(15)+(2√(223))/(3)+(14√(7))/(3)+√(55)) \\ =22.3943\approx22.39 \end{gathered}`

Therefore,

`\int_3^7√(x^2+6)dx\approx22.39`