Question

Math question #1 please show steps

Answer 1

Answer:

C

Step-by-step explanation:

An approximation of an integral is given by:

$\displaystyle \int_a^bf(x)\, dx\approx \sum_{k=1}^nf(x_k)\Delta x\text{ where } \Delta x=\frac{b-a}{n}$

First, find Δx. Our a = 2 and b = 8:

$\displaystyle \Delta x=\frac{8-2}{n}=\frac{6}{n}$

The left endpoint is modeled with:

$x_k=a+\Delta x(k-1)$

And the right endpoint is modeled with:

$x_k=a+\Delta xk$

Since we are using a Left Riemann Sum, we will use the first equation.

Our function is:

$f(x)=\cos(x^2)$

Therefore:

$f(x_k)=\cos((a+\Delta x(k-1))^2)$

By substitution:

$\displaystyle f(x_k)=\cos((2+\frac{6}{n}(k-1))^2)$

Putting it all together:

$\displaystyle \int_2^8\cos(x^2)\, dx\approx \sum_{k=1}^{n}\Big(\cos((2+\frac{6}{n}(k-1))^2)\Big)\frac{6}{n}$

Thus, our answer is C.

*Note: Not sure why they placed the exponent outside the cosine. Perhaps it was a typo. But C will most likely be the correct answer regardless.