Use a net to find the surface area of the cylinder use 3.14 for pi

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

20 POINTS!! EASY QUESTION! WILL MARK BRAINLIEST!! READ THE INFO! THEN FILL IN THE BLANKS!

LUCKY_DIMON [66]

To find the unit rate in cost per hour, you will write the amount of money over the total time for each class. Then divide the cost by the time to find the unit rate.

New Steps: $7.50/1.5 hours = $5 per hour

Stepping High: $6.50/1.25 hours = $5.20 per hour

8 0

2 years ago

Read 2 more answers

Write .0296 in word form

TEA [102]

two hundred ninety-six ten thousandths

4 0

2 years ago

Vertex A in quadrilateral ABCD lies at (-3, 2). If you rotate ABCD 180° clockwise about the origin, what will be the coordinates

natali 33 [55]

The rule for a rotation by 180° about the origin is (x,y) -->(−x,−y)

So A(-3, 2) to A'(3, -2)

Answer:

A'(3, -2)

4 0

2 years ago

What is x, the distance between points A and A'? 2. 4 units 4. 8 units 13. 6 units 14. 4 units.

adelina 88 [10]

The distance between points A (3, -5) and A' (2, -3) is 2.4 units.

Given that,

The points A (3, -5) and A' (2, -3).

We have to determine,

The distance between point A and A'.

According to the question,

The distance between two points is determined by using the distance formula.

$\rm Distance \ formula = \sqrt{(x_2-x_2) ^2+ (y_2-y_2)^2$

Then,

The distance between points A (3, -5) and A' (2, -3) is,

$\rm Distance = \sqrt{(2-3) ^2+ ((-3)-(-5))^2}\\\\ Distance = \sqrt{(-1) ^2+ (2)^2}\\\\ Distance = \sqrt{1+4}\\\\Distance = \sqrt{5}\\\\Distance = 2.4 \ units$

Hence, The distance between points A (3, -5) and A' (2, -3) is 2.4 units.

For more details refer to the link given below.

brainly.com/question/8069952

6 0

1 year ago

Read 2 more answers