Applications of Integrals
This unit is all about the different uses of integrals. It's also very example heavy; to get better, you have try to do as many different types of questions in this unit as possible, so you are aware and ready for all the different types of questions that could come on the test.
We can use integrals to find areas between two curves
suppose we have this graph
well we know that an integral of a function gives us the area between the curve and the x-axis. To find the area between two curves we would just need to take the integrals of the two curves and then subtract getting us the middle region!
so for this we would need the integral of the two functions from the left most point they intersect at to the right most point. This might be obvious on some graphs but others we would have to graph them in the graphing calculator to find the exact points of intersection. Solving for the difference of both the integrals would give us the area we are trying to find.
in the above graph (find graph and complete example)
Okay...BUT what if we spin that 2D area around to get a 3D shape???
I know you guys are super excited to know what happens
When we were taking the area of the 2D object using integrals we were basically taking infinitely small slices of the shape and adding them together to get the area we want. If we took slices of a 3D object that would mean when we add the slices we get a volume.
There are three main methods with which we can solve this kind of problem.
Disk Method:
This method splits the entire object into circular disks and takes the volume of each disk. Each disk is actually a really really thin cylinder which is why we can use the area of base*height formula and sum up all the volumes to get the volume of the entire shape. Thats all we do in disk method. The hard part is writing it in the calculus-y integral form.
Washer Method:
This is one of the methods used when the shape has a hole through it when spun around. This isn't really a distinct method as shell or disk but worth mentioning. This is a modification of disk method. Since there is a hole through every slice we take of the object we have to account for the missing middle area. This is why we take the same method as diisks (using the area of base*height formula) but when we calculate the area we subtract the missing area (which is circular as well) then multiply by height.
Shell Method:
This is arguably the trickiest of the three methods. This method requires looking at a visual model to really understand it. This method essentially takes the entire object and splits into shells that encase each other (I assure you are going to need to look at a model for this).
Average Value Functions
Average means summing up everything then dividing by the number of terms.
We know that the solving for the integral gives us the sum of all the outputs between the bounds of the integral. To find average we now just need to divide by all the integers between the bounds.
That is fraction before the integral is doing. It is doing the division part in finding the average. (This is a VERY recurring problem on the AP Calc AB test; almost every single year has it at least once)