From the very first years of Algebra, to Geometry and Trigonometry, to now Calculus, it's all coming together. In order to keep moving on in the math world you have to retain and keep exercising the principles you learned since 8th grade. It is pretty neat seeing how all of this adds up and how we explore new and efficient ways to do things easier.
This week we started to learn how to find volumes of functions of a solid by using integrals. To help understand and comprehend what is going on, a good visual is strongly recommended. Essentially what ever and where ever you are rotating a visual always helps me picture what the solid would look like.
In the first week, we only examined how to find the volumes of disks and washers. The area of a disk can be found by using this formula: pi(f(x)^2. The are of a washer can be found by using this formula: pi(R^2-r^2). We only focused on these two and these two alone were actually pretty interesting and fun to play/experiment with.
In the area between two curves case we approximated the area using rectangles on each sub interval. For volumes we will use disks on each sub-interval to approximate the area. The same principle can be applied towards the rotation of the y axis. If we were to rotate along the x axis the height then becomes dx, and if likewise along the y axis the height becomes dy.
This week we started to learn how to find volumes of functions of a solid by using integrals. To help understand and comprehend what is going on, a good visual is strongly recommended. Essentially what ever and where ever you are rotating a visual always helps me picture what the solid would look like.
In the first week, we only examined how to find the volumes of disks and washers. The area of a disk can be found by using this formula: pi(f(x)^2. The are of a washer can be found by using this formula: pi(R^2-r^2). We only focused on these two and these two alone were actually pretty interesting and fun to play/experiment with.
In the area between two curves case we approximated the area using rectangles on each sub interval. For volumes we will use disks on each sub-interval to approximate the area. The same principle can be applied towards the rotation of the y axis. If we were to rotate along the x axis the height then becomes dx, and if likewise along the y axis the height becomes dy.