1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
lara [203]
3 years ago
8

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

y equals e Superscript x​, and the line x equals ln 16 about the line x equals ln 16.

Mathematics
1 answer:
Salsk061 [2.6K]3 years ago
7 0

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

You might be interested in
A rectangle measures 2 4/5 inches by 2 4/5 inches. What is its area? Answer in a mixed number. WILL MARK BRAINLIEST
Marysya12 [62]

7 21/25

2 4/5 x 2 4/5 = 7 21/25

7 0
3 years ago
What is 28:36 in simplest form
xeze [42]
It would be 7:9 because if you divide each side by 4 then you can get it in simplest form
6 0
3 years ago
Read 2 more answers
What is 2x(x+5)=4 using the quadratic formula
antoniya [11.8K]
X= \frac{-5(+/=) \sqrt{33} }{2}
Or it would be approximately: x≈ 0.37228132, -5.37228132
Hope this helped! ;P
4 0
3 years ago
The table shows the height of a plant as it grows.
viva [34]

Answer

144

Step-by-step explanation:

12*12

3 0
3 years ago
Karyn proofreads 15 pages in 2 hours for $40.
OLEGan [10]
Hi there.

Since she reads 15 pages in 2 hours, all we have to do to find out how much she reads in 1 hour is divide.

15/2 = 7.5

7.5 pages/hour

Same applies to the pay

40/2 = 20

$20/hour

Karyn proofreading rate is 7.5 pages per hour. Karyn receives on average $20 per hour.

~
8 0
3 years ago
Read 2 more answers
Other questions:
  • Wesley sailed his boat north for 18 miles then due east for another 18 miles approximately how far is Wesley from where he began
    11·2 answers
  • Is the inverse of the function shown below also a function?
    15·1 answer
  • How do you find the median
    7·2 answers
  • Solve lor x/5 + 6 = 10 x=​
    15·1 answer
  • A hat is on sale for h dollars. The regular price is 7 times as much. Charlie has enough money to buy two hats at the regular pr
    11·1 answer
  • I WILL AWARD BRAINLIEST!! PLEASE HELP!!!
    7·2 answers
  • Which of the following is an example of a random event?
    8·2 answers
  • I need this answered quickly pls I’m not sure what the answe is but I have a idea
    8·1 answer
  • Ms. Grayson has budgeted $6500 for a computer lab in her school.
    5·1 answer
  • A team manager buys a team jacket and a team cup for each of the 10 players on the dodgeball team. Each team cup costs $10.70. T
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!