Hello week 16. Friday during class, we were working on a packet. I was getting the answers right! I tried finishing it today, but I started to have problems with the back page. So, I'm going to talk about what I am getting and what I'm not comfortable doing.
Area between curves:
formula: bSa top equation-bottom equation
example: Find the area of the region enclosed by y=-4x^2+41x+94 and y=x-2 between x=1 and x=7.
1. draw a picture: This would be a parabola facing down with a line going through it.
2. subtract the two equations: (-4x^2+41x+94-(x-2)
(-4x^2+41x+94-(-x+2)
3. put together like terms and integrate: S(-4x^2+41x+94-(-x+2))
(-4/3)x^3+20x^2+96x
4. plug in 1 and 6 and subtract: (3584/3)-(344/3)= 1080
Volume by disks:
formula: pi[R(x)]^2dx
example: Find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 4. square root of(-x^2+6x+27)
1. draw a picture: it would be like a circle once you reflect it
2. plug into formula: square root of(-x^2+6x+27)
pi(0S4)[square root of(-x^2+6x+27)]^2
pi(0S4)(-x^2+6x+27)
3. integrate: pi((-1/3)x^3+3x^2+27x)
4. plug in 0 and 4: pi(404/3)-0= (404/3)pi
Please Help Now :)
Now for Washers..? I am confused a little on this. Is there any problems in the packet that need this or another example anyone can show me? I just need help understanding why and when you use that formula.
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you use washers when you are given two equations in the problem. After you realize it's a washer problem, it's really easy. All you ahve to do is graph both of the equations and then see which one is higher, or shows up at the top of the graph. That is your "top" equation and then you plug it into the formula which is
ReplyDeleteS (from # to #) (top)^2 - (bottom)^2
PS. area is the same thing, except the equation isn't squared.
Then you just solve like a disc problem!
washers:
ReplyDeletethe formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
Washers are used with holes. You have holes when you have two equations. The formula is simple. It is basically the same thing as the area between two curves, except each function is squared before you take the definite integral on your interval. Then you just times it by pi and voila, you have volume.
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