Answer:
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
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}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brcl%7DA%20%26%3D%26%202%5Cpi%20%5Cln16%20%5Cint_%7B0%7D%5E%7B%5Cln16%7De%5E%7Bx%7Ddx%5C%5C%5C%5C%26%3D%26%202%5Cpi%5Cln16%5Be%5E%7Bx%7D%5D_%7B0%7D%5E%7B%5Cln16%7D%5C%5C%5C%5C%26%3D%26%202%5Cpi%5Cln16%2816%20-%201%29%5C%5C%26%3D%20%26%5Cmathbf%7B30%5Cpi%5Cln1%7D%5C%5C%5C%5CB%20%26%3D%262%5Cpi%5Cint_%7B0%7D%5E%7B%5Cln16%7Dxe%5E%7Bx%7Ddx%5C%5C%5Cend%7Barray%7D)
Use the integration by parts formula
![\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 $\large \boxed{\mathbf{2\pi(15 - \ln16)}}$}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brcl%7Duv%20-%20%5Cint%20v%20du%20%26%3D%26%5C%5CB%20%26%3D%26%202%5Cpi%5Bxe%5E%7Bx%7D%5D_%7B0%7D%5E%7B%5Cln16%7D%20-%202%5Cpi%20%5Cint_%7B0%7D%5E%7B%5Cln16%7D%20e%5E%7Bx%7Ddx%5C%5C%5C%5C%26%3D%262%5Cpi%5Bxe%5E%7Bx%7D%5D_%7B0%7D%5E%7B%5Cln16%7D-2%5Cpi%20%5Be%5E%7Bx%7D%5D_%7B0%20%7D%5E%7B%5Cln16%7D%5C%5C%5C%5C%26%3D%20%262%5Cpi%2816%5Cln16-%200%29%20-2%5Cpi%2816%20-%201%29%5C%5C%26%3D%26%20%5Cmathbf%7B32%5Cpi%20%5Cln16%20-30%5Cpi%7D%5C%5CA%20-%20B%20%26%3D%20%2630%5Cpi%5Cln16%20-%2032%5Cpi%20%5Cln16%20%2B30%5Cpi%5C%5C%26%3D%26%2030%5Cpi%20-%202%5Cpi%20%5Cln16%5C%5C%26%3D%26%5Cmathbf%7B2%5Cpi%2815%20-%20%5Cln16%29%7D%5Cend%7Barray%7D%5C%5C%5Ctext%7BThe%20volume%20of%20the%20solid%20is%20%24%5Clarge%20%5Cboxed%7B%5Cmathbf%7B2%5Cpi%2815%20-%20%5Cln16%29%7D%7D%24%7D)
Answer:
13,500
Step-by-step explanation:
x + y = 750
750 x 18 = 13.500
Answer: The ordered pair is (2,3)
Step-by-step explanation:
To solve this problem, we can use substitution. With substitution, we substitute the y in each problem with the equations. In this case, x + 1 can substitute the y in y = -x + 5. Here is what the problem looks like now:
x + 1 = -x + 5
The reason we used substitution is because we assume that these linear equations are equal. We must prove it, though. We can use algebra.
Move the variables to one side and the numerical values to another:
x + x = 5 - 1.
Simplify.
2x = 4
Simplify.
x = 2.
Now, we got our x value, we can substitute it into our problem:
y = 2 + 1
y + -2 + 5
2 + 1 = - 2 + 5 = 3
Our x is 2, and our y is 3. The ordered pair that is a solution is (2,3).
Answer:
Step-by-step explanation:
The distance is always positive and is same as the absolute value of subtraction of the two numbers on the line.
<u>This can be shown as:</u>
- |-3/2 - 6| =
- |-1.5 - 6| =
- |-7.5| =
- 7.5
or
- | 6 - (-3/2)| =
- |6 + 1.5| =
- |7.5| =
- 7.5
Use distributive property
x^2+x-14+(-x+22)i
