A number is decreased by seven. Then, the new number is multiplied by 3 to get an answer of -9. What is the original number?

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve

Salsk061 [2.6K]

Answer:

$\large \boxed{2\pi(15 - \ln16)}$

Step-by-step explanation:

A 90° portion of your volume is shown below.

Let's use the shell method and decompose the solid of revolution into cylindrical shells.

The volume of each cylindrical shell is

dV = 2πrh dx

where

r = ln16 - x

h = eˣ

So

dV = 2π(ln16 - x)eˣdx

$\begin{array}{rcl}V& =& 2\pi \int_{0}^{\ln16}(\ln16 - x)e^{x}dx\\\\&=&2\pi \int_{0}^{\ln16}(\ln 16)e^{x}dx -2\pi \int_{0}^{\ln16} xe^{x}dx\\\\&=& 2\pi \ln16 \int_{0}^{\ln16}e^{x}dx - 2\pi\int_{0}^{\ln16}xe^{x}dx\\\\\end{array}$

Let's integrate the two terms separately.

$\begin{array}{rcl}A &=& 2\pi \ln16 \int_{0}^{\ln16}e^{x}dx\\\\&=& 2\pi\ln16[e^{x}]_{0}^{\ln16}\\\\&=& 2\pi\ln16(16 - 1)\\&= &\mathbf{30\pi\ln1}\\\\B &=&2\pi\int_{0}^{\ln16}xe^{x}dx\\\end{array}$

Use the integration by parts formula

$\int u dv = uv - \int v du$

$\begin{array}{rcl}uv - \int v du &=&\\B &=& 2\pi[xe^{x}]_{0}^{\ln16} - 2\pi \int_{0}^{\ln16} e^{x}dx\\\\&=&2\pi[xe^{x}]_{0}^{\ln16}-2\pi [e^{x}]_{0 }^{\ln16}\\\\&= &2\pi(16\ln16- 0) -2\pi(16 - 1)\\&=& \mathbf{32\pi \ln16 -30\pi}\\A - B &= &30\pi\ln16 - 32\pi \ln16 +30\pi\\&=& 30\pi - 2\pi \ln16\\&=&\mathbf{2\pi(15 - \ln16)}\end{array}\\\text{The volume of the solid is $\large \boxed{\mathbf{2\pi(15 - \ln16)}}$}$

7 0

3 years ago

Find each measure Please help me

Kruka [31]

Answer:

1. m∠CGB=120

3. m∠AGD=90

5. m∠CGD=150

2. m∠BGE=60

4. m∠DGE=30

6. m∠AGE=120

Step-by-step explanation:

Sorry that they are out of order.

3 0

2 years ago

Read 2 more answers

What is the decimal equivalent of ⅞? * 4 points 0.780 0.870 0.875 0.885

Lubov Fominskaja [6]

Answer: 0.875

Step-by-step explanation: How I got this answer is I used Division, (7 ÷ 8= 0.875)

Hope this helps, Have a great day/night!

~IFoundJiminsLostJams~

4 0

2 years ago

Read 2 more answers

Pls help helpful answer

zalisa [80]

Answer:

as you can see in both triangles, comparing to each other, x = 90 angle

w = 52, because when 90 + 38 + w = 180 for a triangle

z=10yd

u = 62yd

v = 79 yd

please, also make sure these answers are correct from anyone, but ihave answer correct :)

Hope it helps

4 0

2 years ago

Simplify

melomori [17]

Answer: $4\sqrt{3}$

Step-by-step explanation:

For this exercise it is important to remember the following:

$i=\sqrt{-1} \\\\i^2=-1$

Given the following expression:

$-2i\sqrt{-12}$

You can notice that the radicand (the number inside the square root) is negative. Therefore, in order to simplify the expression, you need to follow these steps:

1. Replace $\sqrt{-1}$ with $i$ and simplify: