Let R be the region enclosed by the x-axis, the graph y = x2, and the line x = 4. Write an integral expression for the volume of the solid whose base is R and whose slices perpendicular to the x-axis are semi-circles.
Answer
The region R is the same as in the example:
This solid is like the one in the example except that instead of the slices being squares, they're semicircles. The slice at position x is a semi-circle with diameter
y = x2 and thickness Δ x.
This solid really looks like a loaf of French bread. Since the diameter of a slice is x2, the radius is . The volume of a slice is the area of the semi-circle (half the area of a circle) multiplied by the thickness, or
The variable x goes from 0 to 4 in this region, so when we take the integral we get
Example 2
Let R be the region enclosed by the x-axis, the graph y = x2, and the line x = 4. Write an integral expression for the volume of the solid whose base is R and whose slices perpendicular to the y-axis are squares.
Answer
The region R is the same as in the example:
Now we're slicing the other way. It's important to get the right base region. R is this:
Not this:
When we cut a slice perpendicular to the y-axis, the slice will look like this:
Slices closer to the x-axis will be bigger and slices farther from the x-axis will be smaller. The whole solid will look like this:
Since the slice at position y is a square, its volume will be the area of the square multiplied by its thickness Δ y. The length of the side of the square is
The volume of the slice is
Since y goes from 0 to 42 = 16 in the region R, these are the limits of integration for the integral. The volume of the solid is
Example 3
Let R be the region enclosed by the x-axis, the graph y = x2, and the line x = 4. Write an integral expression for the volume of the solid whose base is R and whose slices perpendicular to the y-axis are equilateral triangles.
Answer
The region R is the same as in the example:
We're slicing perpendicular to the y-axis again. The base of a slice still has length , but now the slice is an equilateral triangle instead of a square. Our volume looks like a Toblerone.
The area of an equilateral triangle with side-length s is
so the volume of a slice is
The variable y goes from 0 to 16 in the base region R, so the volume of the solid is
Example 4
Let R be the region bounded by y = x and y = x2. Write an integral expression for the volume of the solid with base R whose slices perpendicular to the y-axis are semi-circles.
Answer
R is this region:
The diameter of the semicircular slice at position y is the distance from the line y = x to the curve y = x2, which is .
This means the radius of the semi-circle is
The area of the semi-circle is
so the volume of the slice is
The variable y goes from 0 to 1 in the region R, so the volume of the entire solid is
Example 5
Let R be the region bounded by y = x and y = x2. Write an integral expression for the volume of the solid with base R whose slices perpendicular to the y-axis are squares
Answer
R is this region:
We're slicing the same way before, but now instead of the slice at position y being a semi-circle with diameter , it's a square with side-length .
The volume of the slice at position y will be the area of the square, multiplied by Δy:
Since y still goes from 0 to 1, the volume of the whole solid is
Example 6
Let R be the region bounded by y = x and y = x2. Write an integral expression for the volume of the solid with base R whose slices perpendicular to the x-axis are equilateral triangles
Answer
R is this region:
Now we're slicing the other direction, perpendicular to the x-axis. The slices are equilateral triangles. The side length of the equilateral triangle at position x is the distance from the curve y = x2 to the line y = x. This distance is
x – x2., The area of the equilateral triangle is
so the volume of the slice is
The variable x moves from 0 to 1 in the region R, so the volume of the whole solid is
Example 7
Let R be the region bounded by x2 + y2 = 1. Write an integral expression for the volume of the solid with base R whose slices perpendicular to the y-axis are semi-circles.
Answer
The region R is the unit circle:
If slices perpendicular to the y-axis are semi-circles, we have another loaf of French bread. The radius of the semi-circle at height y is x, where x is the distance from the line x = 0 to the curve x2 + y2 = 1.
Rearranging the equation, we get
The area of the semi-circle at height y is
so the volume of the slice is
The variable y goes from -1 to 1 in the region R, so the volume of the solid is
Alternately, since the solid is symmetric we could find the volume of its upper half and then multiply by 2. This gives us the expression
Example 8
Let R be the region bounded by x2 + y2 = 1. Write an integral expression for the volume of the solid with base R whose slices perpendicular to the x-axis are equilateral triangles.
Answer
The region R is the unit circle:
For this solid we slice perpendicular to the x-axis and get equilateral triangles. The triangle at position x has side-length 2y where :
This means the area of the triangle is
.
The volume of the slice is
and the volume of the entire solid is
Alternately, since the solid is symmetric we could find the area of half of it and then multiply by 2.
In this case, we get the expression
Example 9
Let R be the region bounded by x2 + y2 = 1. Write an integral expression for the volume of the solid with base R whose slices perpendicular to the x-axis are squares.
Answer
The region R is the unit circle:
For this solid we slice perpendicular to the x-axis and get squares. The slice at position x has side-length .
The volume of this slice is
The volume of the entire solid is
We could also find the volume by calculating the volume of half the solid and then multiplying by 2.
Then we get the integral expression
For symmetric objects like the one above, it's easier to find the volume by finding the volume of half the solid and then multiplying by 2. It's pretty likely you'll be asked to finish the problem by evaluating the integral, and it's easier to evaluate an integral if one of the endpoints is 0.
. The volume of a slice is the area of the semi-circle (half the area of a circle) multiplied by the thickness, or
, but now the slice is an equilateral triangle instead of a square. Our volume looks like a Toblerone.
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, it's a square with side-length 
.
:
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