Monday - no school
Tuesday - B-rob wasn't there; studied for limits test
Wednesday - took limits test and started on integration
Integration is the area under a curve. We learned four different ways of approximating this area. These four ways included four riemann sums--approximations of area using rectangles or trapezoids. These four ways are:
LRAM - left hand approximations
x[ f(a) + f(a +x) + ... f(b) ]
RRAM - right hand approximations
x[ f(a + x) + ... f(b) ]
MRAM - midpoint approximations
x[ f(mid) + f(mid) + ... ]
Trapezoid - approximations using a trapezoid
x/2[ f(a) + 2f(a + x) + 2f(a + 2x) + ... f(b) ]
MRAM and Trapezoid are the most precise of the approximations, though MRAM is a pain. These are very easy to do. To begin, one must find x. To find x = b-a/n, then plug into equations.
Ex: find the area of x - 3 on the interval [0,2] n = 4
1. x = 2-0/4 = 1/2
2. 1/2[ f(0) + f(1/2) + f(1) + f(3/2) ]
*Remember, the number of terms should match up with your n
3. 1/2[ -3 - (5/2) - 2 - (3/2) ] = 1/2 - 9 = -9/2
*The answer may come out negative, but remember it's area that you're finding, and area cannot be negative, so your answer will become positive.
Wednesday - I don't remember?
Thursday - instead of approximating areas, we learned to find the direct answers
There are two types of integration: indefinite and definite. Indefinite integration is when the answer comes out to be an equation, and definite integration is when the answer comes out to be a number.
An integral is basically a backwards derivative. For integration, all the same properties of derivitives apply, except you do the opposite. Instead of subtracting 1 from the exponent and multiplying it to the term, you will add 1 to the exponent and divide the term by it.
For definite integrals: integral a b = F(b) - F(a) = #
I pretty much understand indefinite integrals, but I'm having trouble on the homework on the ones with definite integrals. I'm getting mixed up when there is a product or quotent rule. Also, I remember in class we learned something about splitting the integral up and adding the two split up parts together? When do we do that?
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Just remember that there is no product or quotent rules, period. you're just messing with your exponents.
ReplyDeleteas far aw splitting the intergral up and adding the two.. we did that and we only do that when there is an absolute value in the problem
You split it up when there's an absolute value and when there's something like (x+2)/x....I THINK! You might want to ask someone else though...
ReplyDeleteyes, you do split it up when there is an absolute value. im not sure about the fraction thing though.
ReplyDeleteanyways, whenever you have something in a fraction that you would usually use quotient rule for.. you simply just make the fraction a negative exponent. for example:
(x+2)/x^2
this would become
(x+2)(x^-2)
then you would just integrate it from there. just remember that you never use product and quotient rule with integrals. and don't forget the + c :-)