Question

Use the Trapezoid Rule to approximate the value of the definite integral

Answer 1

Can u help no one is helping me ;(

Answer 2

7.0625

Step-by-step explanation:

The trapezoidal rule ( this is an approximation ) tells you that the average of the left and right endpoints should be as follows,

`\int _a^bf\left(x\right)dx\:\approx (\Delta \:x)/(2)\left(f\left(x_0\right)+2f\left(x_1\right)+...+2f\left(x_(n-1)\right)+f\left(x_n\right)\right)`

where
`\Delta \:x\:=\:(b-a)/(n)` ... at this point we can apply the Riemann Formula, in order to divide the interval 0 ≤ x ≤ 2 into n = 4 subintervals of length
`\:\Delta x=(1)/(2)\:`.

`x_0=0,\:x_1=(1)/(2),\:x_2=1,\:x_3=(3)/(2),\:x_4=2` ,

`(\Delta x)/(2)=((1)/(2))/(2)=(1)/(4)`

=
`(1)/(4)\left(f\left(x_0\right)+2f\left(x_1\right)+2f\left(x_2\right)+2f\left(x_3\right)+f\left(x_4\right)\right)` - Let's calculate the sub intervals for each, substituting to receive our solution.

`f\left(x_0\right)= 0` ( this is as 0⁴ is 0 )

`2f\left(x_1\right)= 1/8` ( this is as
`2\left((1)/(2)\right)^4=1/8` )

`2f\left(x_2\right)=2` ( 2
`*` 1⁴ is 2 )

`2f\left(x_3\right)= 81/8` ( this is as
`2\left((3)/(2)\right)^4 = 81/8` )

And finally
`f\left(x_4\right) = 16`, as 2⁴ is 16. Therefore, let us plug in our solutions for each into the primary expression, and solve,

`(1)/(4)\left(0+(1)/(8)+2+(81)/(8)+16\right)` = 7.0625 - this is our solution. The correct answer is option c, and i hope this clarifies why.