Area, Volume, and Arc Length Exercises

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

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

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

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

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

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

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

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

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.