Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #32 : How To Find Volume Of A Region

Sphere segment

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 2 units to the right of the origin and the second cut is 5 units to the right of the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the sphere might be projected onto the two-dimensional Cartesian plane:

Sphere segment outlined

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 #40 : How To Find Volume Of A Region

A solid is formed by rotating the region bound by , , and  about the y-axis.

Find the volume of this solid. 

Possible Answers:

Correct answer:

Explanation:

In order to find the volume of this solid, the solid must be formed and observed. 

Problem 11

This solid represents the solid that is formed by taking half a parabola (xbound by x = 0) and giving it a height of 9 units (bounded by y = 9). This solid forms an object that resembles a vase. The volume of the solid would then be an integral of the area of the cross sections of the vase. Since the vertical cross sections (cutting the vase with vertical planes) give unsymmetrical shapes, it would be better to use horizontal cross sections and change the orientation of the figure to better see how to set up the volume integral.

Problem 11a

This orientation can be obtained by changing the axis. This is accomplished by make x the dependent variable, and y the independent variable. To do this, solve the equation for the function in term of x instead of y:

Now the area of the cross-sections must be calculated. Sample cross sections are shown in the figure above. Since the area of a circle is equal to πr2, the differential area expression for this solid would be:

, now we have the area of the cross section as a function of y

The final step is to find the limits of integration, which in this case, would be from the bottom of the solid to the top. Since it starts at y = 0, and continues to y = 9, those are the limits of integration. To find the volume:

 

 

Example Question #41 : How To Find Volume Of A Region

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 2 and a major axes(the horizontal length) of 10, 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?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

Example Question #42 : Volume

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 1 and a major axes(the horizontal length) of 12, has a segment with perpendicular planes cut out of it, such as is pictured above.

If the first cut is 5 units to the left of the origin and the second cut is 4 units to the left of the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

Example Question #43 : Volume

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 2 and a major axes(the horizontal length) of 6, 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 2 units to the right of the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

Example Question #44 : Volume

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 1 and a major axes(the horizontal length) of 50, 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 14 units to the right of the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

Example Question #45 : Volume

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 8 and a major axes(the horizontal length) of 20, has a segment with perpendicular planes cut out of it, such as is pictured above.

If the first cut is at the origin and the second cut is 5 units to the right of the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

Example Question #46 : Volume

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 2 and a major axes(the horizontal length) of 5, 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 left of the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

Example Question #47 : Volume

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 1 and a major axes (the horizontal length) of 3, 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 at the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

Example Question #48 : Volume

Ellipse segment

A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of  and a major axes (the horizontal length) of, has a segment with perpendicular planes cut out of it, such as is pictured above.

If the first cut is  units to the left of the origin and the second cut is  units to the right of the origin, what is the volume of the segment?

Possible Answers:

Correct answer:

Explanation:

To approach this problem, begin by picturing how the spheroid might be projected onto the two-dimensional Cartesian plane:

Ellipse segment outlined

In its two-dimensional projection, we notice the outline of an ellipse. The formula for an ellipse with horizontal and vertical axes,  and  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 the ellipse contacts the x-axis, the greatest and smallest possible values of  on the ellipse:

This is how the volume of a prolate/oblate spheroid can be derived, and similary, this is how we can derive the area of our segment. We're told that our equatorial radius . and the length of the spheroid , or :

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