Question

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

Answer 1

Answer:

7.0625

Explanation:

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

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

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

$x_0=0,\:x_1=\frac{1}{2},\:x_2=1,\:x_3=\frac{3}{2},\:x_4=2$ ,

$\frac{\Delta x}{2}=\frac{\frac{1}{2}}{2}=\frac{1}{4}$

= $\frac{1}{4}\left(f\left(x_0\right)+2f\left(x_1\right)+2f\left(x_2\right)+2f\left(x_3\right)+f\left(x_4\right)\right)$ - 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 $2\left(\frac{1}{2}\right)^4=1/8$ )

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

$2f\left(x_3\right)= 81/8$ ( this is as $2\left(\frac{3}{2}\right)^4 = 81/8$ )

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,

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

Answer 2

Can u help no one is helping me ;(