Prompt: Please describe how you would approach a known cross section problem that you haven’t done before (like when the sections are trapezoids for examples). What is your approach if you can’t cut paper to help you visualize the solid that will be created? How did this activity help you understand finding volumes with known cross sections? Why might this be useful in the real world?
If I approached a cross section with a shape that I haven't done before I would first figure out how to find the area of each cross section. So if it were a trapezoid I would find the area of the trapezoid, if it were an octagon then an octagon, etc. Then I would substitute f(g) g(x), whatever it is in replacement for the values in the formula. Then I would integrate what I have over the interval. Done.
If I can't cut paper to visualize a problem and I'll try to visualize it in my head, but if I'm still having trouble grasping what is going on then I'll draw a hideous rendition of the shape that is going to be made. That would help me picture what is going on quite a bit.
This activity helped me a lot in visualizing what is going on when we are doing these sorts of problems, and when I can tell what's going on it is also easier to grasp how to solve the problem, and why what we do to solve the problem works.
In the real world this concept could be utilized in finding the volume of shapes - duh. It can also be used to optimize a shape's form, or to know how to make a solid fit inside something else or fit together with something else.
If I approached a cross section with a shape that I haven't done before I would first figure out how to find the area of each cross section. So if it were a trapezoid I would find the area of the trapezoid, if it were an octagon then an octagon, etc. Then I would substitute f(g) g(x), whatever it is in replacement for the values in the formula. Then I would integrate what I have over the interval. Done.
If I can't cut paper to visualize a problem and I'll try to visualize it in my head, but if I'm still having trouble grasping what is going on then I'll draw a hideous rendition of the shape that is going to be made. That would help me picture what is going on quite a bit.
This activity helped me a lot in visualizing what is going on when we are doing these sorts of problems, and when I can tell what's going on it is also easier to grasp how to solve the problem, and why what we do to solve the problem works.
In the real world this concept could be utilized in finding the volume of shapes - duh. It can also be used to optimize a shape's form, or to know how to make a solid fit inside something else or fit together with something else.