A community for students. Sign up today!
Here's the question you clicked on:
 0 viewing
 2 years ago
Let R be the planar region between the curves y^2 + (x1)^2 = 4 and y^2 = 3(x1) which contains the point (0, 0). 1.) Calculate the total area of R. 2.) Determine the total length of the boundary of R.
 2 years ago
Let R be the planar region between the curves y^2 + (x1)^2 = 4 and y^2 = 3(x1) which contains the point (0, 0). 1.) Calculate the total area of R. 2.) Determine the total length of the boundary of R.

This Question is Closed

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0Here's the plot, x going from 1 to 3, y from (squareroot(3)) to squareroot(3)

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1354694787271:dw

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0??? Isn't R the region in between?

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1354694904341:dw

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1354694942463:dw

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0I still have no idea how to calculate the total area or boundary

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2so the region is the area of a circle , minus that parabola bit

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2can you find the radius of the circle

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2so the area of the circle is ...

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2maybe some integration

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0that's exactly what I need, would love to learn how

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2have you found the intersection points ?

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0intersections at (2, squareroot(3)), and (2, squareroot(3))

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1354696097101:dw

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2lets calculate this area dw:1354696370978:dw

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2we can find the area of this region by integrating dw:1354696620853:dw \[=\int\limits_{x=1}^{x=2}\int\limits_{y=0}^{y=\sqrt{3(x1)}}\text dy\text dx\] \[=\int\limits_{y=0}^{y=\sqrt{3}}\int\limits_{x=\tfrac{y^2}3+1 }^{x=2}\text dx\text dy\]

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.2maybe there is any easier way im not seeing

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1youre doing ok, you just seem to be cutting it up way to much

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1y^2 + (x1)^2 = 4 and y^2 = 3(x1) move it so that the circle is centered at the origin; the area stay the same, we just move it y^2 + (x1+1)^2 = 4 and y^2 = 3(x1+1) y^2 + x^2 = 4 and y^2 = 3x x = sqrt(2^2y^2) and x = 1/3 y^2 and it really doesnt matter if we have this oreiented long the x or y soo if its simpler to play with as y = sqrt(2^2x^2) and y = 1/3 x^2 thats fine too

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1so, what does all this moving and rewrite do? dw:1354733679539:dw it just makes it easier to see the equation to integrate with

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1the area to remove is between the circle and parabola \[\int_{a}^{b}\sqrt{4x^2}\frac13x^2~dx\]

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0Thanks! What about the total length of the boundary of R?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1hardest part might be in determining the length of the parabolas arc, circles are a dime a dozen

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1do you recall any details about how to work a line integral?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1354757262089:dw

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1\[\int_{C}ds=\int_{a}^{b}\sqrt{(x')^2+(y' )^2}\]

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1since x = x x' = 1 y = 1/3 x^2 y' = 2/3 x \[\int_{a}^{b}\sqrt{(1)^2+(\frac23x )^2}~dx\]

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0that integral equals the boundary?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1the integral equals the length of the curve of the parabola im sure you can figure out the circle part

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1354758192335:dw

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1find that and times 2 for the circle part

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1354758715223:dw

Merciless
 2 years ago
Best ResponseYou've already chosen the best response.0so 2x 1 + the integral of the curve (4.4) and I'd have 6.4 as total boundary length?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1\[y=\sqrt{4x^2}\] \[y'=\frac{x}{\sqrt{4x^2}}\] \[\int_{2}^{\sqrt3}\sqrt{1+\frac{x^2}{4x^2}}dx\]

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1circle part is, already did the parabola \[2\int_{2}^{\sqrt3}\sqrt{1+\frac{x^2}{4x^2}}dx\] \[2\int_{2}^{\sqrt3}\sqrt{\frac{4}{4x^2}}dx\] \[2\int_{2}^{\sqrt3}\frac{2}{\sqrt{{4x^2}}}dx\] \[dx=2sin(u)~:~dx=2cos(u) du\] \[2\int_{2}^{\sqrt3}\frac{2}{\sqrt{{44sin^2(u)}}}2cos(u) du\] \[2\int_{2}^{\sqrt3}\frac{2}{2\sqrt{{1sin^2(u)}}}2cos(u) du\] \[2\int_{2}^{\sqrt3}\frac{2}{2cos^2(u)}2cos(u) du\] \[2\int_{2}^{\sqrt3}\frac{2}{cos(u)} du\] \[4\int_{2}^{\sqrt3}sec(u) du\] \[4(ln(sec(u)+tan(u))\]and either adjust back to xs, or adujst the limits in terms of us
Ask your own question
Ask a QuestionFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.