All Calculus 1 Resources
Example Questions
Example Question #2904 :Functions
Find the volume of the functionrevolved around the-axis on the interval.
cubic units
cubic units
cubic units
cubic units
cubic units
The formula for the volume of the region revolved around the x-axis on the intervalis given as
where.
As such,
.
And by the corollary of the First Fundamental Theorem of Calculus
.
As such, the volume is
cubic units.
Example Question #2905 :Functions
Find the volume generated by rotating the functionaround the设在在间隔.
To approach this problem, treat the functionas the radius of a disc, with infintesimal thickness dx (which has the x-axis going through its center).
This infinitesismal volume of this disc is:
Now the volume of the solid rotated around the x-axis is the sum of these discs. In other word, it is the integral:
Note that the derivative ofis, so from this we can infer that the integral ofis:
Example Question #2906 :Functions
Find the volume of the solid generated by rotating the parabolaaround the-axis, over the region where the function is greater than or equal to zero.
In stating the bounds, the question is asking that we use those values of x where the function intercepts the x-axis:
Now, treat the functionas the radius of a disc with infinitesimal thickness. The volume of this disc is thus
And the volume can found via integration:
Example Question #2907 :Functions
For this problem assume that volumeis given by:
, whereis the surface area function.
Given the formula for surface area of a structure is given by.
Determine the volume functionof this structure.
For this problem assume thatis not a function of.given below is a constant.
We're given that
Since the indefinite integral is with respect to, we can move the surface area function in front of the integral since we are told thatis not a function of.
Using the power rule,
, whereis a constant.
In our problem
Therefore, volume can be given as
Example Question #2908 :Functions
Using the method of cylindrical disks, find the volume of the region of the graph of the function
on the interval.
units cubed
units cubed
units cubed
units cubed
units cubed
The formula for volume is given as
where.
As such,
.
When taking the integral, we will use the inverse power rule which states
.
Applying this rule term by term we get
And by the corollary of the first Fundamental Theorem of Calculus,
As such, the volume is
units cubed.
Example Question #2909 :Functions
Using the method of cylindrical disks, find the volume of the region of the graph of the function
on the interval.
units cubed
units cubed
units cubed
units cubed
units cubed
The formula for volume is given as
where.
As such,
.
When taking the integral, we will use the inverse power rule which states
.
Applying this rule term by term we get
.
And by the corollary of the first Fundamental Theorem of Calculus,
.
As such, the volume is
units cubed.
Example Question #2910 :Functions
Set up the integral to find the volume of the solid if the region is bounded byand, revolved about.
For this problem, we will use the shell method. Write the formula for the shell method.
The thickness of the shell is.
The radius of the shell is the distance from the y-axis to the vertical rectangles,.
The height of the shell is the top curve subtract the bottom curve. The top curve isand the bottom curve is. The height is:.
Determine the bounds by setting the top and bottom equations equal to each other.
Solve forroots.
The roots areand will be the bounds to the integral. Write the integral.
Example Question #21 :Volume
A sphere with a radius of 5 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 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 radiusin the Cartesian coordinate system is given as:
Which can be rewritten in terms ofas
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:
whereand
The integral sums up these thin cylinders to give the volume of the shape.
Treatingas our, this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values ofon 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 #22 :Volume
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 radiusin the Cartesian coordinate system is given as:
Which can be rewritten in terms ofas
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:
whereand
The integral sums up these thin cylinders to give the volume of the shape.
Treatingas our, this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values ofon 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 #23 :Volume
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 radiusin the Cartesian coordinate system is given as:
Which can be rewritten in terms ofas
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:
whereand
The integral sums up these thin cylinders to give the volume of the shape.
Treatingas our, this integral can be written as:
Consider the points where circle contacts the x-axis, the greatest and smallest possible values ofon 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: