wow, post 32, we have been in school forreevvverrr... but anyway, the school year is almost over and this is just gonna be one less blog i have to do.
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.
since we have to post things we have trouble with, so other people can comment and help us out, i am going to go ahead and say that i suck at integrating things with ln, e, or sin and cos n stuff, but after doing that packet on integration, i can do the simple integrating, just not the stuff i mentioned above.
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The integral of sin is just -cos since the derivative of -cos is sin and
ReplyDeletethe integral of cos is sin since the derivative of sin is cos
Example: Integrate: cos (2x)
So you know the integral of cos now is sin, but you are left with a 2.
To get rid of the 2, you have to multiply the integral by 1/2.
The integral of cos (2x) = 1/2 sin (2x) + c
You can check your answer by taking the derivative of that
The derivative of 1/2 sin(2x) is 1/2 cos (2x) (2) or cos (2x)
Another common integral with sin and cos is
Integrate cos x/ sin x
Remember that if the top of the fraction is the derivative of the bottom, it is natural log integration.
The derivative of sin x is cos x.
Therefore the integral is ln |sin x| + c
Integrate the (the square root of x - 1) ^2/ the square root of x
ReplyDeleteu= the square root of x -1 du = 1/2 x^-1/2
since there is a negative exponent, the x^-1/2 will go to the bottom becoming x^1/2, which is what is needed
but there is an extra 2 at the bottom. To get rid of the 2, you will have to multiply by 2
2 integral u^2
2(1/3) u ^3
2/3 u^3
2/3 (the square root of x -1)^3 +C
You know something is a natural log when the top is the derivative of the bottom.
ReplyDeleteFor e integration, the exponent of e is always u. Once you find u, do substitution like normal except you'd integrate e^u du.