volume+of+a+solid+with+known+cross-sections

1. Look at this diagram of a rather strange solid. It has a base that is defined by y = √x over the interval [0, 9], but each cross-section is a semi-circle! [|cross sections are semicircles]

[|another view, using "slabs"]

Suppose you were to slice this solid and find the area of each cross section.

a. Explore: suppose we want to look at the slab that stretches from x = 4 to x = 4.001. **What is the thickness of this slab**?

whichever is easier. It would help to draw this semi-circle. If this is situated at x = 4, what is the diameter of the semi-circle? What is the radius? We will consider that the slab is a prism, and use the volume formula V = Bh. **What is the volume of this slab**?
 * What is the area of the "base"?** (i.e. the cross section, which happens to be a semi-circle). You can either use x = 4 or x = 4.001,

b. Now let's find the volume of the slab that stretches from x = k to x = k + dk. **What is the thickness of this slab**? x = k. **What is the volume of this slab?**
 * What is the area of the "base"?** (i.e. the cross section; the semicircle!). It would definitely help to draw this semicircle. You really should use

c. Note that there are multiple slabs, each restricted by the function y = √x and all having cross sections that are semicircles. What is the sum of the volumes of all of such slabs defined by x values stretching from x = 0 to x = 9?

d. What would be the volume of the solid defined as above, but with x-values stretching from x = 4 to x = 9?

e. What would be the volume of the solid defined as above, but with x-values stretching from x =8 to x = 9? Give a good argument for your answer being reasonable.

2. In the following solid, the base is defined by the circle x²+ y² = 1, but we are limited to the first quadrant.

[|base is a quarter-circle, cross sections are squares]

a. Explore: We might slice this solid into slabs that are perpendicular to the x-axis. Suppose we want to look at the slab that stretches from x = 0.5 to x = .5001. **What is the thickness of this slab**?

whichever is easier. It would help to draw this square. If this is situated at x = 0.5, what is the length of a side of the square? What is the area of the square?
 * What is the area of the "base"?** (i.e. the cross section, which happens to be a square). You can either use x = 0.5 or x = .5001,

We will consider that the slab is a prism, and use the volume formula V = Bh. **What is the volume of this slab**?

b. Now find the volume of a random slab, located at "x" and with thickness "dx." What is the length of a side of this square, in terms of x?
 * What is the area of this cross section (square, in this case)**?


 * What is the volume of this slab**?

c. What is the sum of the volumes of all such slabs defined by x - values stretching from x = 0 to x = 1?

3. Next one: [|base is a circle, cross sections are isosceles right triangles!!]

a. Let's slice this solid into slabs that are perpendicular to the x-axis. What is the thickness of each slab?

b. Since we know that the base is a circle with radius 2, what is the equation for the base? Solve this for y.

c. We know that each cross-section is an isosceles right triangle. Which part of the right triangle lies on the circle?

d. Draw a cross-section at a random value of x, and label the sides of the triangle with what you know. What is the area of this triangle, in terms of x?

e. What is the volume of each slab? What is the sum of the volumes of all such slabs defined by x-values stretching from x = -2 to x = 2?