Question

Approximate the area of the shaded region by using the Trapezoidal Rule with n=4.

Answer 1

Answer:1075

Given the shaded region, you need to use the Trapezoidal Rule with

In order to approximate the area of the shaded region.

Use points: (-20,20), (-10,30), (0,40), (10,20), and (20,15)

Answer 2

Given the shaded region, you need to use the Trapezoidal Rule with:

`n=4`

In order to approximate the area of the shaded region.

By definition, the Trapezoidal Rule is:

`T_n\approx\int_a^bf(x)dx=(\Delta x)/(2)\lbrack f(x_0)+2f(x_1)+...f(x_(n-1))+f(x_n)`

You need to find:

`\Delta x`

This can be found with this formula:

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

Notice in the graph that, in this case, the interval is:

`\lbrack-20,20\rbrack`

Therefore:

`\begin{gathered} a=-20 \\ b=20 \end{gathered}`

Then, by substituting values into the formula, you get:

`\Delta x=(20-(-20))/(4)=(40)/(4)=10`

In this case, you can determine that the subintervals must begin at -20 con adding 10 until you get to 20:

`\begin{gathered} x_0=-20 \\ \\ x_1=-10 \\ \\ x_2=0 \\ \\ x_3=10 \\ \\ x_4=20 \end{gathered}`

Substitute values into the Trapezoidal Rule Formula:

`T_4\approx(10)/(2)\lbrack f(-20)+2f(-10)+2f(0)+2f(10)+f(20)\rbrack`

You need to use the graph to identify the y-values that correspond to each x-value. Identify the points on the curve with those coordinates:

`(-20,20),(-10,30),(0,40),(10,20),(20,16)`

Therefore, by substituting the corresponding y-values into the formula and evaluating, you get:

`T_4\approx(10)/(2)\lbrack20+2(30)+2(40)+2(20)+16\rbrack`
`T_4\approx1080`

Hence, the answer is:

`1080`