Divide each one out to see which is cheaper. 4.39/12 = .37 cents per pint while 2.89/8 = .36 cents per pint. These are cost per unit problems.The second choice is the cheaper of 2 by a cent.

4 0

3 years ago

The region in the first quadrant bounded by the x-axis, the line x = ln(π), and the curve y = sin(e^x) is rotated about the x-ax

charle [14.2K]

First, it would be good to know that the area bounded by the curve and the x-axis is convergent to begin with.

$\displaystyle\int_{-\infty}^{\ln\pi}\sin(e^x)\,\mathrm dx$

Let $u=e^x$ , so that $\mathrm dx=\dfrac{\mathrm du}u$ , and the integral is equivalent to

$\displaystyle\int_{u=0}^{u=\pi}\frac{\sin u}u\,\mathrm du$

The integrand is continuous everywhere except $u=0$ , but that's okay because we have $\lim\limits_{u\to0^+}\frac{\sin u}u=1$ . This means the integral is convergent - great! (Moreover, there's a special function designed to handle this sort of integral, aptly named the "sine integral function".)

Now, to compute the volume. Via the disk method, we have a volume given by the integral

$\displaystyle\pi\int_{-\infty}^{\ln\pi}\sin^2(e^x)\,\mathrm dx$

By the same substitution as before, we can write this as

$\displaystyle\pi\int_0^\pi\frac{\sin^2u}u\,\mathrm du$

The half-angle identity for sine allows us to rewrite as

$\displaystyle\pi\int_0^\pi\frac{1-\cos2u}{2u}\,\mathrm du$

and replacing $v=2u$ , $\dfrac{\mathrm dv}2=\mathrm du$ , we have

$\displaystyle\frac\pi2\int_0^{2\pi}\frac{1-\cos v}v\,\mathrm dv$

Like the previous, this require a special function in order to express it in a closed form. You would find that its value is

$\dfrac\pi2(\gamma-\mbox{Ci}(2\pi)+\ln(2\pi))$

where $\gamma$ is the Euler-Mascheroni constant and $\mbox{Ci}$ denotes the cosine integral function.

5 0

3 years ago

You are ordering a pizza and soda for the big game. A deep dish pizza costs $10.25, and a two-liter of soda is $2.00. For each t

Step2247 [10]

Add 10.25 and 2.00 keep subtractin the value you got with 1.50 until you get to your answer....Please mark brainliest...

8 0

3 years ago