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
mina [271]
1 year ago
10

Calculus 3 help please.​

Mathematics
1 answer:
Reptile [31]1 year ago
8 0

I assume each path C is oriented positively/counterclockwise.

(a) Parameterize C by

\begin{cases} x(t) = 4\cos(t) \\ y(t) = 4\sin(t)\end{cases} \implies \begin{cases} x'(t) = -4\sin(t) \\ y'(t) = 4\cos(t) \end{cases}

with -\frac\pi2\le t\le\frac\pi2. Then the line element is

ds = \sqrt{x'(t)^2 + y'(t)^2} \, dt = \sqrt{16(\sin^2(t)+\cos^2(t))} \, dt = 4\,dt

and the integral reduces to

\displaystyle \int_C xy^4 \, ds = \int_{-\pi/2}^{\pi/2} (4\cos(t)) (4\sin(t))^4 (4\,dt) = 4^6 \int_{-\pi/2}^{\pi/2} \cos(t) \sin^4(t) \, dt

The integrand is symmetric about t=0, so

\displaystyle 4^6 \int_{-\pi/2}^{\pi/2} \cos(t) \sin^4(t) \, dt = 2^{13} \int_0^{\pi/2} \cos(t) \sin^4(t) \,dt

Substitute u=\sin(t) and du=\cos(t)\,dt. Then we get

\displaystyle 2^{13} \int_0^{\pi/2} \cos(t) \sin^4(t) \, dt = 2^{13} \int_0^1 u^4 \, du = \frac{2^{13}}5 (1^5 - 0^5) = \boxed{\frac{8192}5}

(b) Parameterize C by

\begin{cases} x(t) = 2(1-t) + 5t = 3t - 2 \\ y(t) = 0(1-t) + 4t = 4t \end{cases} \implies \begin{cases} x'(t) = 3 \\ y'(t) = 4 \end{cases}

with 0\le t\le1. Then

ds = \sqrt{3^2+4^2} \, dt = 5\,dt

and

\displaystyle \int_C x e^y \, ds = \int_0^1 (3t-2) e^{4t} (5\,dt) = 5 \int_0^1 (3t - 2) e^{4t} \, dt

Integrate by parts with

u = 3t-2 \implies du = 3\,dt \\\\ dv = e^{4t} \, dt \implies v = \frac14 e^{4t}

\displaystyle \int u\,dv = uv - \int v\,du

\implies \displaystyle 5 \int_0^1 (3t-2) e^{4t} \,dt = \frac54 (3t-2) e^{4t} \bigg|_{t=0}^{t=1} - \frac{15}4 \int_0^1 e^{4t} \,dt \\\\ ~~~~~~~~ = \frac54 (e^4 + 2) - \frac{15}{16} e^{4t} \bigg|_{t=0}^{t=1} \\\\ ~~~~~~~~ = \frac54 (e^4 + 2) - \frac{15}{16} (e^4 - 1) = \boxed{\frac{5e^4 + 55}{16}}

(c) Parameterize C by

\begin{cases} x(t) = 3(1-t)+t = -2t+3 \\ y(t) = (1-t)+2t = t+1 \\ z(t) = 2(1-t)+5t = 3t+2 \end{cases} \implies \begin{cases} x'(t) = -2 \\ y'(t) = 1 \\ z'(t) = 3 \end{cases}

with 0\le t\le1. Then

ds = \sqrt{(-2)^2 + 1^2 + 3^2} \, dt = \sqrt{14} \, dt

and

\displaystyle \int_C y^2 z \, ds = \int_0^1 (t+1)^2 (3t+2) \left(\sqrt{14}\,ds\right) \\\\ ~~~~~~~~ = \sqrt{14} \int_0^1 \left(3t^3 + 8t^2 + 7t + 2\right) \, dt \\\\ ~~~~~~~~ = \sqrt{14} \left(\frac34 t^4 + \frac83 t^3 + \frac72 t^2 + 2t\right) \bigg|_{t=0}^{t=1} \\\\ ~~~~~~~~ = \sqrt{14} \left(\frac34 + \frac83 + \frac72 + 2\right) = \boxed{\frac{107\sqrt{14}}{12}}

You might be interested in
100 students in a classroom like iPhone or Samsung mobile or both mobile. Out kf them, 40 both mobiles. It the ratio of number o
ladessa [460]

The number of students who like Iphone: 46

I hope that answers your question!

4 0
2 years ago
The graph below shows the solution set of which inequality? 
lakkis [162]
I hope this helps you

6 0
3 years ago
A human body that is performing light work generates about 650 BTUs of body heat per hour. If a storage room that is 20 ft × 25
Damm [24]

in case your wodering its 2.1 BTUs/ft^3

6 0
3 years ago
Please help me with this No LINKS and please explain to get brainliest and also no inappropriate answers
Alex787 [66]

Answer:

Range: 22

Median: 79

Mode: 80

Step-by-step explanation:

I hope this helped and have a nice day! Also can I have brainliest?

3 0
2 years ago
a Car has a 15 gallon gas tank . I f a car gets 32 miles per gallon and travels 384 miles . how much gas will be left in the tan
Komok [63]

Answer:

so you would actually multiply 32 x 15=480 then subtract 480 - 384= it would have 96 miles left. then divide 96/32= 3

so your answer would be 3 gallons left

Step-by-step explanation:

4 0
2 years ago
Other questions:
  • Expand and simplify (x-2)(x-2)(x-3)
    15·1 answer
  • What is 2.1 with 1repeating as a mixed number
    15·1 answer
  • What formula is used to determine which fraction is the larger one
    13·1 answer
  • Part A. At the local baker you can get 10 cookies for $2. How many cookies could you get for $5 dollars?
    11·1 answer
  • Th upper-left coordinated on a rectangle are (3, -3) and the upper-right coordinates are (7 -3). The rectangle has an area of 16
    15·1 answer
  • Given: △OKE∼△VDE<br> Prove: EK/EO = ED/EV
    6·1 answer
  • Polygon ABCD is defined by the points A(-4,2) B(-2, 4), C(1, 3) &amp; D(2,2). Match the coordinates of the points of the transfo
    7·1 answer
  • A company can send a maximum of 40 workers to complete a
    15·1 answer
  • Question 6
    13·1 answer
  • Find the area of the surface generated by revolving the curve about the axis y = 9-x^2
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!