Zoklet.net I need calculus help...
 User Name Remember Me? Password
 Register Home IRC Chat FAQ Members List Social Groups - View All Groups - Your Group Messages Calendar Search Today's Posts Mark Forums Read

 Advertisement SECURITY TAG DETACHERS FREE shipping, FREE sample tags FREE 24/7 live tech support vinipooh123@gmail.com Advertisement Download torrents anonymously No logs - Anonymous IP www.portalvpn.com
#1
05-19-2009, 03:01 AM
 Giggles The Panda Member Join Date: Jan 2009 Location: Bumblefuck, Egypt Thanks: 6 Thanked 2 Times in 2 Posts
I need calculus help...

The tangent lines to the parabola y=ax^2+b at the point (2,3) and (-2,3) contain the origin.

(a) Find the value of a and b.
(b) Find the area of the region R enclosed by the parabola and the two tangent lines.
(c) Set up and evaluate an integral expression in one variable which gives the volume when the region R, described in part (b), is revolved about the y-axis.

Please show your work and maybe give a brief explanation of what you did.
#2
05-19-2009, 03:34 PM
 Mathematics Peasant Join Date: Jan 2009 Location: UK Thanks: 11 Thanked 41 Times in 29 Posts
Re: I need calculus help...

Quote:
 Originally Posted by Giggles The Panda The tangent lines to the parabola y=ax^2+b at the point (2,3) and (-2,3) contain the origin. (a) Find the value of a and b. (b) Find the area of the region R enclosed by the parabola and the two tangent lines. (c) Set up and evaluate an integral expression in one variable which gives the volume when the region R, described in part (b), is revolved about the y-axis. Please show your work and maybe give a brief explanation of what you did.
(a) Find the value of a and b.

The form of both lines is:

$y=mx,$

as there is no constant (the lines pass throught the origin). The gradients of the lines are then given by dy/dx:

$y=\frac{3}{2}x$

and

$y=-\frac{3}{2}x.$

We now know what the gradient of the curve must be at these points, so we can differentiate and find the constant a:

$y=ax^2+b,$

$\frac{\partial y}{\partial x}=2ax.$

Substituting in the gradients and the values of x:

$2 a (2)=3/2$

$2 a (-2)=-3/2$

$a=3/8.$

So we have:

$y=\frac{3}{8}x^2+b.$

From the lines we also know the values of y at x=2 and x=-2, so we can find b:

$3=\frac{3}{8}2^2+b,$

$b=3/2.$

(b) Find the area of the region R enclosed by the parabola and the two tangent lines.

The basic stratergy is to integrate between 0 and x=2 (where the line touches the curve), take away the area between the line and the x-axis (just a triangle) and double the answer, as the curves are symmetric in the y-axis.

$\int _0^2 \left(\frac{3}{8}x{}^2+\frac{3}{2}\right) dx=4$

(Click "Show steps" for details: http://www81.wolframalpha.com/input/?i=Integrate+3%2F8x^2%2B3%2F2)

Area between line and x-axis:

$\frac{1}{2}\left(base \times height \right) = \frac{1}{2}(2)(3)=3$

Area under curve - area of triange = 1

So total area enclosed by the curve and the two lines = 2*1 = 2.
 The following users say "It is so good to hear it!": Giggles The Panda (05-20-2009)
#3
05-22-2009, 12:12 AM
 Mathematics Peasant Join Date: Jan 2009 Location: UK Thanks: 11 Thanked 41 Times in 29 Posts
Re: I need calculus help...

Quote:
 Originally Posted by Giggles The Panda (c) Set up and evaluate an integral expression in one variable which gives the volume when the region R, described in part (b), is revolved about the y-axis.
Volume of revolution when a function is rotated around the y-axis:

$V=\int \pi x^2 dy$

Notice that the function has been changed so it is in terms of y i.e. instead of writing y=... you are re-arranging and writing x=... (function of y).

If you imagine the volume swept out by rotating the line y=3/2x around the y-axis between y=0 and y=3 you will have a cone of volume:

$V_1=\int_{0}^3 \pi \left(\frac{2}{3}y\right)^2 dy=4\pi$

You can then simply subtract the volume swept out by the quadratic as it rotates. The quadratic doesn't touch the x-axis; at x=0 y=3/2. So you need to integrate between y=3/2 and y=3.

$V_2=\int_{3/2}^3 \pi \left(\frac{8}{3} \left(y-\frac{3}{2}\right) \right)dy=3\pi$

$V_R=V_1-V_2=\pi$

 Bookmarks

 Tags calculus

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home Official Bullshit     News         From M&A     Support and Suggestions         Rejected         Completed General     Generally Speaking         What's Good     Pissin' Each Other Off     Bat Country     The Trading Pit Entertainment     The Book Nook         Original Works     Fragyard         Games Poor People Play     LOL, Internet         I need to find it on the internet         Pics, or it didn't happen     Music!         Albums of 2012 Discussion         Albums of 2013 Discussion         The Musician's Playground     TV and Movies         Everything Anime     Visual Arts     Viva La Sports Hobbies     Bad Ideas         Revenge, Vandalism & Lulzy Bad Ideas     DIY     Gearheads     There's a Pyro in All of Us Human Life     Alternative Living     Better Living Through Chemistry         From Seed to Smoke         I'm Gonna Kick Tomorrow     Carved Outta Wood         So Last Summer         Holes and Ink     Gluttons & Gourmets     Graduation to Obligation     Love, Lust, and Relationships     Neurons & Needles     This Just In! Science     Flasks and Beakers     Flora, Fauna, and Green Living         Outdoor Living     Math, Science, and Engineering Society     The Inhumane Condition         History     Religion and Spirituality     Der Politik     Paranormal     It's a Conspiracy!     Weapons and Combat     Mission Support     Travel and World Language Technology     Technophiles and Technophiliacs         Codes of all kinds             Zokbot                 Suggestions     Network Security & Hacking Other     The Trashcan

 Similar Threads Thread Thread Starter Forum Replies Last Post Hyper-dimension Math, Science, and Engineering 9 04-26-2009 06:02 PM boostinkyboo Math, Science, and Engineering 14 02-24-2009 04:43 AM

All times are GMT. The time now is 04:39 PM.

 Hot Topics How did you pick what religion you are? Fucked over? Oil question more of that teen mom porn from mtv What do you know a lot about and when/where did you start learning? who wins in a fight? EDC spyderco ebay paypal advice Anxiety over angry people One of those dayz How to cash out a hacked/stolen paypal account ? How To Avoid Detection Growing Weed Indoors "White Girl's Club" busted up by jew's minions. Girls sent to re-education. Does Science Refute God? Where to buy CC's for carding? Belgium waffles On IRC #fragyard Users: 4 Messages/minute: 0 Topic: "http://www.zoklet.net/..." #totse Users: 24 Messages/minute: 0 Topic: "go team!" #zok Users: 10 Messages/minute: 0 Topic: "vaginaboob" Advertisements Your ad could go right HERE! Contact us!
 -- Parent ---- NewSchool ---- Sunshine ---- Dark / Red ---- Nostalgia ---- Green ---- NotGreen ---- lol pink ---- Industrial / Yellow -- Mobile - Contact Us - Zoklet.net Forums - Archive - Top