The more variety the better Or drinking this:
This is indeed a chest of teaI often forget exactly how much of each of these items I consume. Cheesecake is too heavenly to bother with thinking about the amount of the decadent treat that you are eating. Tea is... well TEA. The only thing that people drink more of is WATER and its the national drink of ENGLAND. Civilized peoples drink this and civilized peoples certainly don't think about how much they drink.
Now that we have the random facts and foodstuffs out of the way, on to the Calculus.
So... What does an amnesic experience with food consumption have to do with Calculus?
Now that we have the random facts and foodstuffs out of the way, on to the Calculus.
So... What does an amnesic experience with food consumption have to do with Calculus?
INTEGRALS
Otherwise known as the "antiderivative"
The use of an integral is (you guessed it) opposite of the use of a derivative. Instead of finding a rate of change from a function of how much of "something" there is at a point in time, you can use an integral to find exactly how much of "something" there is during a certain period of time.
Yes, this would be the other half of Calculus, and like everything else in Mathworld such as addition / subtraction, multiplication / subtraction, and e / ln, the integral is indeed the opposite of a derivative.
The integral is defined as the area under the curve, which looks something like this:
As indicated, when the area of the curve is above the x axis, then the integral is positive and when it is below the x axis, then it it negative.
The integral is defined as the area under the curve, which looks something like this:
The use of an integral is (you guessed it) opposite of the use of a derivative. Instead of finding a rate of change from a function of how much of "something" there is at a point in time, you can use an integral to find exactly how much of "something" there is during a certain period of time.
Now, yet again there are certain shortcuts that you can use whilst doing integrals.
One of the most used and most common is:
One of the most used and most common is:
REVERSE POWER RULE!
As you (hopefully) have guessed by now, since the integral is the opposite of a derivative then you can apply the opposite of the power rule in order to do integrals.
Oh those mathematicians... Such creative names...
As you (hopefully) have guessed by now, since the integral is the opposite of a derivative then you can apply the opposite of the power rule in order to do integrals.
This means that you add one to the exponent of each term and then divide the term by the new exponent. This can be represented by:
There are actually two types of integrals. The one shown above is called an indefinite integral. This means that instead of getting a numerical answer, we will get an expression. This is similar to taking the derivative of a function without establishing what x equals first.
NOTE: The "dx" means with respect to 'x'. If the variable you wanted to integrate was 'y', then you would write "dy" instead.
Then there's the problem of the big "C" in the middle of the page. It stands for "constant" and indicates that when you take an integral of something you are missing some information.
Remember that when you take a derivative of a function that has a constant of it, you lose the constant because the power rule causes the constant to be multiplied by 0:
Essentially, if you just had 2 as a constant, then it would look something like the above. Since x^0 = 1, then 2x^0 = 2. And if you apply the power rule when taking the derivative then it would equal 0.
And so, it is apparent here that some information is lost here. The all important "+2" disappears into oblivion.
Which is why when you take the integral:
You have to add the "+C" there to indicate that there may be a constant missing.
Then we move on to the second type of integral... If you had to guess what it is called based on the fact that the first type of integral is indefinite integral... Yes, it is indeed a definite integral.
A definite integral varies from an indefinite integral in that a definite integral will get you a definite answer...
The point of an integral is to get the area under the curve, or change from a rate of change of "something" into how much of "something" over an area or period of time.
In order to do this, "bounds" have to be set. These bounds indicate the period in which you want to know how much of "something" there is.
An example of how the process works looks like this:
The bottom generally contains the lower bound and the top contains the higher bound of the area you want to integrate. If this is not the case for a problem, then your answer will have to be multiplied by "-1".
First, you take the integral of your function and then you use a vertical bar to indicate what your bounds are. Then you plus in your upper bound to the integral and subtract the lower bound plugged into the integral.
Questions? Email me at Jonathan352@gmail.com or comment.
An integral practice problem will be posted soon in order to clear some things up. Oh, and this time it will have cheesecake and a certain historical figure...
NOTE: The "dx" means with respect to 'x'. If the variable you wanted to integrate was 'y', then you would write "dy" instead.
Then there's the problem of the big "C" in the middle of the page. It stands for "constant" and indicates that when you take an integral of something you are missing some information.
Remember that when you take a derivative of a function that has a constant of it, you lose the constant because the power rule causes the constant to be multiplied by 0:
Essentially, if you just had 2 as a constant, then it would look something like the above. Since x^0 = 1, then 2x^0 = 2. And if you apply the power rule when taking the derivative then it would equal 0.
Which is why when you take the integral:
Then we move on to the second type of integral... If you had to guess what it is called based on the fact that the first type of integral is indefinite integral... Yes, it is indeed a definite integral.
A definite integral varies from an indefinite integral in that a definite integral will get you a definite answer...
The point of an integral is to get the area under the curve, or change from a rate of change of "something" into how much of "something" over an area or period of time.
In order to do this, "bounds" have to be set. These bounds indicate the period in which you want to know how much of "something" there is.
An example of how the process works looks like this:
First, you take the integral of your function and then you use a vertical bar to indicate what your bounds are. Then you plus in your upper bound to the integral and subtract the lower bound plugged into the integral.
Questions? Email me at Jonathan352@gmail.com or comment.
An integral practice problem will be posted soon in order to clear some things up. Oh, and this time it will have cheesecake and a certain historical figure...
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