Ok, so my internet is finally up and I can type up and post the blog that I have had written down for days... so for this blog i'm going to go over volume and area of disks. a disk is a solid object, and to find the volume of it, you just solve for x or y of the given equation. you can tell by the problem which axis its about. so you draw the equation, then you reflect it. the formula is S r^2 dx, and the radius is the equation that is given. and the formula for the area of a disk is the same thing but the radius is not squared.
Example: Find the volume of the solid obtained by rotating about the x-axis the region under the curve from -2 to 1: y=sqroot(-2x^2-10x+48)
so, since the formula needs to you to square the equation, the square root just disappears. then, you have to integrate the equation, then you do top-bottom, and multiple by pi. so after you integrate it looks like this: -2/3(x)^3-5x^2+48x
then after you plug in everything and do top-bottom it looks like this: -2/3(1)^3-5(1)^2+48(1)-(-2/3(-2)^3-5(-2)^2+48(-2))
then you can simplify it on your own even more, or plug it all into your calculator. i chose to simplify it more by hand first because with that many numbers it is pretty easy to make a mistake in your calculator. So i further simplified it to -(2/3)-5+48+(16/3)+20+96 which equals: 491/3.
so then, you have to multiply that by pi, like i mentioned above, and the final answer is 491(pi)/3.
One thing I struggle with is related rates, if anyone has any tips for that I guess I could use them.
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ok so related rates:
ReplyDeletefirst thing you wanna do for that, is pick out everything that is given to you. write it all down on the side.
next thing you wanna do is find out what you are looking for. for ex, if they say what is the rate at which the volume of the sphere is changing. you would know ok, well i need the volume of a sphere formula.
so write that down, & then since it said that you were looking for the RATE @ WHICH IT'S CHANGING, you need to take the derivative of that formula. so dV/dt is what you are looking for.
then simply plug all of the stuff that is already given to you, into your formula & solve for dV/dt
Related Rates:
ReplyDelete1: identify all variables in equations
2: identify what you are looking for
3: sketch and label
4: write an equation involving your variables. (you can only have one unknown so a secondary equation may be given)
5: take the derivative with respect to time.
6: substitute derivative and solve.