Let W be part of the solid sphere of radius 4, centered at the origin, that lies above the plane z = 2. Set up an integral in cylindrical coordinates to find the volume of the region. Do NOT actually integrate.
Answers
Answer:
V = ∫∫∫rdrdθdz integrating from z = 2 to z = 4, r = 0 to √(16 - z²) and θ = 0 to 2π
Step-by-step explanation:
Since we have the radius of the sphere R = 4, we have R² = r² + z² where r = radius of cylinder in z-plane and z = height² of cylinder.
So, r = √(R² - z²)
r = √(4² - z²)
r = √(16 - z²)
Since the region is above the plane z = 2, we integrate z from z = 2 to z = R = 4
Our volume integral in cylindrical coordinates is thus
V = ∫∫∫rdrdθdz integrating from z = 2 to z = 4, r = 0 to √(16 - z²) and θ = 0 to 2π
Answer:
[tex]\displaystyle V = \int\limits^{2 \pi}_{0} \int\limits^{2\sqrt{3}}_{0} \int\limits^{\sqrt{16 - r^2}}_2 {r} \, dz \, dr \, d\theta[/tex]
General Formulas and Concepts:
Multivariable Calculus
Cylindrical Coordinate Conversions:
[tex]\displaystyle x = r \cos \theta[/tex][tex]\displaystyle y = r \sin \theta[/tex][tex]\displaystyle z = z[/tex][tex]\displaystyle r^2 = x^2 + y^2[/tex][tex]\displaystyle \tan \theta = \frac{y}{x}[/tex]Volume Formula [Cylindrical Coordinates]:
[tex]\displaystyle V = \iiint_T \, dV \rightarrow V = \iiint_T {r} \, dz \, dr \, d\theta[/tex]
Step-by-step explanation:
Step 1: Define
Identify given.
Solid sphere w/ radius 4 centered at (0, 0, 0) → x² + y² + z² = 4²
Plane z = 2
Step 2: Find Volume Pt. 1
Find bounds of z.
[Sphere] Substitute in plane z:[tex]\displaystyle x^2 + y^2 + 2^2 = 4^2[/tex]Substitute in cylindrical conversions:
[tex]\displaystyle (r \cos \theta)^2 + (r \sin \theta)^2 + 2^2 = 4^2[/tex]Simplify:
[tex]\displaystyle r^2 + 4 = 16[/tex]Solve for r:
[tex]\displaystyle r = 2\sqrt{3}[/tex]
∴ 0 ≤ r ≤ 2√3
Step 2: Find Volume Pt. 2
[Sphere] Substitute in plane z:[tex]\displaystyle x^2 + y^2 + 4 = 16[/tex]Isolate variables:
[tex]\displaystyle x^2 + y^2 = 12[/tex][Unit Circle] Graph [See 2nd Attachment]
∴ 0 ≤ θ ≤ 2π
Step 3: Find Volume Pt. 3
[Sphere] Substitute in cylindrical conversions:[tex]\displaystyle (r \cos \theta)^2 + (r \sin \theta)^2 + z^2 = 4^2[/tex]Simplify:
[tex]\displaystyle r^2 + z^2 = 4^2[/tex]Solve for z:
[tex]\displaystyle z = \sqrt{16 - r^2}[/tex]
∴ 2 ≤ z ≤ √(16 - r²)
Step 4: Find Volume Pt. 4
Substitute in variables [Volume Formula - Cylindrical Coordinates]:[tex]\displaystyle V = \int\limits^{2 \pi}_{0} \int\limits^{2\sqrt{3}}_{0} \int\limits^{\sqrt{16 - r^2}}_2 {r} \, dz \, dr \, d\theta[/tex]
∴ the volume of the region is found with the integral above.
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Learn more about multivariable calculus: https://brainly.com/question/17203772
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Topic: Multivariable Calculus
Unit: Triple Integral Applications
Related Questions
Assume the random variable X is normally distributed with mean μ = 50 and standard deviation σ= 7. Compute the probability.
Answers
Complete question is;
Assume the random variable X is normally distributed with mean μ = 50 and standard deviation σ = 7. What is the probability P (X > 42)?
Answer:
P(X > 42) = 0.1271
Step-by-step explanation:
We are given;
mean; μ = 50
Standard deviation; σ = 7
Formula for the z-score is;
z = (x - μ)/σ
Thus;
z = (42 - 50)/7
z = -1.14
Since we are looking for P (X > 42), then let's look up this z-value from the z-distribution table attached.
We have;
P(X > 42) = 0.1271
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[tex]\large\mathfrak{{\pmb{\underline{\red{Step-by-step\:explanation}}{\red{:}}}}}[/tex]
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Evaluate/Combine like terms.
[tex] \large{2x + {x}^{2} - 5}[/tex]
This step is optional but it's the best to arrange the degree.
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(f+g)(x) = x²+2x-5Let me know if you have any doubts!
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Answer:
10.A
11.A
12.A
Step-by-step explanation:
thank me later
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Answer:
Option(A)
Step-by-step explanation:
y=4x+2
When you replace the value of x in the equation you get the value of y
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the answer is A since if we lay it out we will get y
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2=0*4+2
6=1*4+2
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Explanation:
1. divide the numerator by the denominator
2. write down the whole number result
3. use the remainder as the new numerator
Answer:
6 1/6
Step-by-step explanation:
Divide the numerator (19) by the denominator (6)
Write down the whole number result which in this case would be 3
Use the remainder as the new numerator over the denominator. This is the fraction part of the mixed number. which in this case is 1/6
therefore the answer is 6 1/6
In parallelogram DEFG if m FGD=125 find m GDE
Answers
9514 1404 393
Answer:
55°
Step-by-step explanation:
Adjacent angles in a parallelogram are supplementary.
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Answer:
Option 4) -$45 per month; the amount of money withdrawn per month.
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Answer:
Step-by-step explanation:
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What is radioactive isotope?
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Here, mass decreases by 73% every hour.
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According to question,
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Answer:
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=====================================================
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Step-by-step explanation:
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Answers
Answer:
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Step-by-step explanation:
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He can draw a straight line from the origin that passes through (0, 0.75).
He can draw a straight line from (3, 0.75) that passes through (6, 2).
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Answers
the graph that he needs to do is a straight line from the origin that passes through (3, 0.75), the correct option is the last one.
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Assuming this is a proportional relation of the form:
y = k*x
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If the number of orders at a production center this month is a Geom(0.7) random variable, find the probability that we'll have at most 3 orders.
Answers
Answer: 0.973
Step-by-step explanation:
The probability mass function of geometric distribution is
[tex]f(x)=(1-p)^{x-1}p,\ \ \ x=1,2,3,...[/tex], p= probability of success on a trial.
As per given, we have
p=0.7
The probability that we'll have at most 3 orders [tex]P(X\leq3)=P(x=1)+P(x=2)+P(x=3)[/tex]
[tex]=(1-0.7)^{1-1}(0.7)+(1-0.7)^{2-1}(0.7)+(1-0.7)^{3-1}(0.7)\\\\=0.7+0.21+0.063\\\\=0.973[/tex]
hence, the required probability = 0.973
The probability that we'll have at most 3 orders will be "0.973".
According to the question,
→ [tex]x \sim Geometric (p)[/tex]
here,
p = 0.7
then,
→ [tex]P(X=x) = (1-p)^{x-1}\times p[/tex]
By putting the given values, we get
[tex]= (1-0.7)^{x-1}\times 0.7[/tex]
hence,
The probability that we'll have at most 3 orders will be:
→ [tex]P(X \leq 3)=P(X=1)+P(X=2)+P(X=3)[/tex]
[tex]=0.7+0.21+0.063[/tex]
[tex]=0.973[/tex]
Thus the above response is correct.
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What is the area of the parallelogram below?
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Answers
Answer:
14 (B)
Step-by-step explanation:
Area = L * h
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h = 2
Area = 7 * 2
Area = 14 in^2
Answer:
14
Step-by-step explanation:
Did lesson
What is an opposite number?
Answers
Answer:
The opposite of a number is the number on the other side of 0 number line, and the same distance from 0.
