24 inches per minute converts to how many miles per day

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

4 years ago

Add, Subtract, Multiply , or Diviide? ANSWER ASAP 100 points

Mama L [17]

Answer:

Add

Step-by-step explanation:

When multiplying exponents with same base the operation is adding up

a^b*a^c = a^(b+c)

8 0

3 years ago

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Andrew deposits 300 into an account that earns 2% interest each year.After the first year,Andrew has 306 in the account. After t

zubka84 [21]

Answer:

idk

Step-by-step explanation:

idk

6 0

3 years ago

14.4 A Book and a Cake A book costs $45. Sales tax on the book is 7%. Write two different expressions that represent the final c

snow_tiger [21]

Answer: 1. $45*1.07 | $45(1.07) 2. (0.85)*32 represent the final price of the cake in dollars because (0.85) is the 15% off in decimals.

Step-by-step explanation:

1 if you multiply 45 by 1.07 it will be 48.15 and that's the final cost of the book, same with $45(1.07) is just another way to represent the multiplication.

2 As I said the 0.85 represent the 15% off, and when you multiply it will be less.

3 0

3 years ago

The distance from the center of a round table top to the edge of the table top is 4 ft. What's the area of the table top?

son4ous [18]

C would be the answer because to find the area of a circle A=pi times radius squared and you put that in a calculator and get 50.24

8 0

4 years ago

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