Hello once again everyone! Time for another guide; this one is going to cover integration. This part of the course can get a little nasty, lots of strange little tricks and surprises. But usually, integration is actually not too bad, as long as you are careful and, you guessed it, have had lots of practice. That way none of the tricks will be new, you'll know what to do, and you'll laugh at BOSTES and their pathetic attempts to confuse you. So let's take a look at some examples. Remember, if anything is unclear, pop a question below, it's very quick to
register. And be sure to check out the
awesome free notes available for integration and other subjects. The
integration notes are fantastic, and will arm you with a university level knowledge of integration. A set of integrals you should know is listed there also.
Before anything, I think it's essential to have a grasp of what integration is. Now this is more uni level stuff, but integration is a geometric thing, linked to limits, partitions and Riemann sums, which allows us to find the area under a curve. It is actually different than an antiderivative, but it just so happens that the process is the exact same! You should have an understanding of how integrals actually work, beyond the basic rules. I won't cover that here, but
let me know below if there is interest for it.
You should know that there are two types of integrals, definite and indefinite. Indefinite integrals are easier, but the HSC markers continually say, people forget the constant of integration! You don't have to know the mathematical reason why it has to be there, but
please remember to write it! Lets look at an example of an indefinite integral.
Example 1 (2014 HSC): Find 
A simple, one step solution. Use these sorts of questions to brush up on your integration rules.
Definite integrals are a little trickier, but no constant of integration! Just remember the formula:
dx=F(B)-F(A) } )
Example 2 (2014 HSC):
Keep your working clear, and remember the formula above, and these questions become easy marks (though this particular interval is a bit tricky, it's worth 3 marks).
There are two main applications of integrals and antiderivatives in 2 unit. Generic geometry questions could be asked, or velocity/acceleration questions. So be sure to practice those. Also asked almost every year, however, is an area under the curve question. These can be troublesome. Let's look at the process to get you set.
Example 3 (HSC 2014): The parabola 
and the line 
intersect at the origin and at x=3. Find the area enclosed by the line and the curve. I'll note that the original question asked us to find the second point of intersection (x=3), done easily with simultaneous equations. What they did give us is a diagram.
This is my biggest piece of advice for area questions: Draw a picture . If you don't, I don't care how good you are, it is really easy to make a mistake. It takes 10 seconds, if they don't give you one, draw it
Remember that the area between two curves (or in this case, a curve and a line) is just the area of the curve on top minus the area of the curve on the bottom. And we can even merge the intervals! So let's do that:
dx\\ =\int _{0}^{3}(-2x^2+6xdx\\ ={\left[\frac{-2}{3}{x}^{3}+3{x}^{2}\right]}_{0}^{3}\\ =(-18+27)-0\\ =\quad 9{ u }^{ 2 } )
There are lots of tricks to be careful of in these sort of questions. And don't be optimistic, they WILL try and trick you.
1- Make sure that, if any areas are under the x axis, that you take the absolute value of the definite integral. If you don't, the area will end up negative, so if this happens, this is the likely culprit.
2- Be prepared to take areas relative to the y axis, not just the x axis. This is done simply by rearranging to make y the subject of the function, and performing the integration as you would normally do.
3- The trick above, that the area enclosed between two curves is just the one on top minus the one on the bottom, can be used no matter where the curves are, above or below the x axis. It technically isn't mathematically correct, but if you were to separate everything and do it properly, you would get the same answer. Try it, it's kind of cool. You won't get penalised, so seize the shortcut!
But mainly, my tip is, be careful. The tricks are easy to spot once you know what they are. It's almost always a sign thing.
The last big part of 2U integration is volumes of revolution. These are normally simple uses of formula such as below:
Example 4 (2014): The region bounded by the curve and the x-axis between x = 0 and x = 4 is rotated about the x-axis to form a solid. Find the volume of the solid.
}^{2}dx}\\\quad=\pi\int_{0}^{4}{(1+2\sqrt{x}+x)dx}\\\quad=\pi{\left[x+\frac{4}{3}{x}^{\frac{3}{2}}+\frac{{x}^{2}}{2}\right]}_{0}^{4}\\\quad=\pi\left[4+\frac{32}{3}+8\right]\\\quad=\frac{68\pi}{3}{u}^{3})
Just remember the formula
and it will get you through most questions you will face, the only complication they could add is doing in respect to the y axis instead of the x axis. Rearrange and it works exactly the same.
But wait, there's one more thing. Simpsons and trapezoidal rules. They are used to estimate areas under curves, and there is almost always a question on one of the two (I'd say 99% chance). There are a bunch of forms of the equation, all very similar, so just stick with what you were taught. I won't cover a question, but instead remind you of how each one works. The Trapezoidal Rule draws straight lines between each point in question, and takes areas of trapezoids. The Simpson's Rule draws a parabola through the points and takes the area under that curve as a Riemann Sum. You may get asked which is better in a specific situation. There is an easy way to tell; draw the straight lines through each point, like a dot to dot, then draw a parabola as smooth as you can through the points. Which gives a better estimate to the area (which is closer to the actual curve)?
Extension 1 students have one more thing to deal with, substitution.
Spoiler
The great thing about Extension 1 Math is that you get given the substitution (at higher levels you need to make the decision, and it can be more time consuming than the math itself).
Example: Use the substitution below to evaluate the integral below:
Make your working for these questions clear; I like to do all my differentiating outside of the integral.
We then substitute and evaluate. If you read one sentence in this guide, read this one... Remember to substitute the changes to the upper and lower limits of the integral! So many people forget, don't let it be you
}\\=\frac{1}{3}(\tan^{-1}{e}-\frac{\pi}{4}) )
The other thing to remember is that if you are doing an indefinite integral, make sure you put the answer back in terms of x. Definite integrals are a bit easier in the sense that you can forget about x once it is out of the integrand.
That's all folks! Be sure to ask a question below, and/or let me know if there is interest for an explanation of integration. I personally found it really hard to grasp, and it helps you understand what's happening while you do the questions. Have fun
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