All Calculus 1 Resources
Example Questions
Example Question #22 : How To Find Volume Of A Region
A sphere with a radius of 10 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 4 units to the left of the origin and the second cut is 3 units to the right of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #21 : Regions
A sphere with a radius of 3 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 2 units to the left of the origin and the second cut is 1 unit to the right of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #31 : How To Find Volume Of A Region
A sphere with a radius of 7 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 6 units to the left of the origin and the second cut is 5 units to the left of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #31 : Regions
A sphere with a radius of 7 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 3 units to the left of the origin and the second cut is 2 units to the left of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #33 : How To Find Volume Of A Region
A sphere with a radius of 4 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 3 units to the left of the origin and the second cut is 3 units to the right of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #34 : How To Find Volume Of A Region
A sphere with a radius of 12 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 6 units to the left of the origin and the second cut is 3 units to the left of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #35 : How To Find Volume Of A Region
A sphere with a radius of 11 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 3 units to the left of the origin and the second cut is at the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #31 : How To Find Volume Of A Region
A sphere with a radius of 2 has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is 1 unit to the left of the origin and the second cut is 1 unit to the right of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #37 : How To Find Volume Of A Region
A sphere with a radius of six has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is six units to the left of the origin and the second cut is three units to the left of the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment:
Example Question #31 : How To Find Volume Of A Region
A sphere with a radius of six has a segment with perpendicular planes cut out of it, such as is pictured above.
If the first cut is three units to the left of the origin and the second cut is at the origin, what is the volume of the segment?
To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:
In its two-dimensional projection, we notice the outline of a circle. The formula for a circle of radius in the Cartesian coordinate system is given as:
Which can be rewritten in terms of as
Now note the disk method of volume creation wherein we rotate a function around an axis (for instance, the x-axis):
The new function in the integral is akin to the formula of the volume of a cylinder:
where and
The integral sums up these thin cylinders to give the volume of the shape.
Treating as our , this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values of on the circle:
This is how the volume of a sphere can be derived, and similary, this is how we can derive the area of our segment: