Refer to the green function as f(x) and the red function as g(x) as you answer the questions below.
1. Click and drag any portion of the applet so that you have an overhead view shape. What integral would represent the area of the base of the function? Be sure to use f(x) and g(x) in your answer.
2. Adjust the view of the figure so that you can see through the figure from one side to the other. Change the value of t by moving the slider. What changes as you move t?
3. Note that in the warm up, the volume of the prisms could be found by multiplying the area of the base by the height of the prism. How is this shape different than the prisms?
4. In a way you can consider the "base" of this shape an equilateral triangle. [br]a. Describe the length of one side of the equilateral triangle using f(x) and g(x). [br]b. What is the formula for area of an equilateral triangle?[br]c. Using your answer from a, write the formula that you can use to find the area of any red equilateral triangle in the figure.
5. Use the drop down to change the red shape to square. How is this solid different than the solid with equilateral triangles? Feel free to use slider to help you reflect.
6. The "base" of this solid is a square. How can you use f and g to find the length of one edge of the square? How can you then use f and g to find the area of a square in the solid?
7. In isosceles triangles 1 and 2, the height of the red triangle is equal to the base of the red triangle. How could you find the area of the red triangle at any point? Be sure to use f(x) and g(x) as you did in questions 4 and 6.
8. Change the drop down menu so that semicircles are formed. How is this figure different from the other solids? Feel free to move the slider to help you reflect.
9. How can you use f(x) and g(x) to find the radius of the red semicircle? What is the formula for the area of any semicircle in the solid, using f and g?
10. Reflection: We will more formally define how to find the area of these solids in the next lesson. What calculus strategy do you think we will use to find the volume? Does this remind you of any other topics that we have covered so far? What confused you the most in this activity? What do you wonder about the figure?