Calculus ab Help » detect Volume making use of Integration » find Cross-Sections: triangle & Semicircles

Find the volume of the solid who cross-sections room equilateral triangles and whose basic is a decaying of radius

*
.

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*


*


*



Correct answer:


Explanation:

Because the disc is that radius R, the base is identified by the adhering to formula:

*

The correct formula for the area of an it is provided triangle is as follows:

*
, with
*
gift the side size of the triangle.

By applying this formula to our general volume formula

*
, we obtain the following:
*
.

The radius R specifies the bounds as being

*
. Next, s have the right to be uncovered by expertise that the value is the distance from the top to the bottom that the circle at any given suggest along
*
. The size of one next of the it is provided triangle, therefore, is
*
.

Putting it all together, the following is obtained:

*

*Note: the problem did no specify if the overcome sections were perpendicular to the 

*
or
*
axis. Because the base is a circle, this must not adjust the resulting volume. The only distinction should be the use of 
*
or
*
as variables in the correct expression.


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Example inquiry #2 : uncover Cross Sections: triangle & Semicircles


Find the volume that the solid whose base is bounded through the one

*
 and who cross-sections are ideal isosceles triangles perpendicular come the
*
axis, with one leg on the basic of the solid.


Possible Answers:
Correct answer:

*


Explanation:

Because the base is a one of radius

*
, the border are defined as
*
.

The area of a appropriate isosceles triangle have the right to be uncovered using the formula

*
, where
*
is the leg size of the triangle. By applying this come our basic volume formula
*
, we acquire the following:
*

The expression because that

*
deserve to be discovered by expertise the truth that the foot
*
the the triangle is on the basic of the solid. The worth is twice the height of the semicircle 
*
*

Putting it every together, the adhering to is obtained:

*
*


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Example question #3 : uncover Cross Sections: triangles & Semicircles


Identify the correct expression for the volume of the solid who cross-sections space equilateral triangle perpendicular come the 

*
axis and also whose base is bounded through
*
 and
*
.


Possible Answers:
Correct answer:

*


Explanation:

First, the cross sections being perpendicular to the 

*
axis suggests the expression must be in terms of
*

The area the an it is intended triangle is

*
, with 
*
being the side size of the triangle. By applying this formula come our general volume formula (
*
), we gain the following:
*
.

The intersection clues of the functions 

*
and 
*
are 
*
and
*
. The 
*
coordinates of these points will specify the bounds for the integral, since our expression is in regards to
*
.

The base is bounded by 

*
and also
*
. Rewriting these attributes in regards to
*
, the adhering to equations space obtained: 
*
 and
*
. Since 
*
is farther native the 
*
axis, the exactly expression because that the side length is
*
.

Putting it all together, the complying with is obtained:

*


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Example inquiry #4 : discover Cross Sections: triangles & Semicircles


Identify the correct expression because that the volume of the hard whose basic is bounded by

*
,
*
, and
*
, and whose cross-sections are best isosceles triangles, perpendicular come the 
*
axis, with one leg on the basic of the solid.


Possible Answers:
Correct answer:

*


Explanation:

First, the cross sections being perpendicular come the 

*
axis suggests the expression have to be in regards to
*
. The area that a appropriate isosceles triangle deserve to be uncovered using the formula
*
, where 
*
is the leg size of the triangle. By using this come our basic volume formula
*
, we acquire the following:
*
.

The intersection point out of the functions defining the an ar are 

*
and
*
. The 
*
collaborates of this points will specify the bounds for the integral, due to the fact that our expression is in regards to
*
.

The basic is bounded through

*
*
and also
*
. Because the cross-sections room perpendicular come the 
*
axis, the foot of the triangle cross-sections are identified by:
*
.

Putting it every together, the adhering to is obtained:

*


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Example inquiry #5 : uncover Cross Sections: triangle & Semicircles


Find the volume that the solid who cross-sections are equilateral triangles and also whose base is a decaying of radius

*
.


Possible Answers:
Correct answer:

*


Explanation:

Because the disc is the radius

*
, the basic is defined by the adhering to formula:
*

The exactly formula for the area of an it is intended triangle is together follows:

*
, through s gift the side length of the triangle.

By applying this formula to our general volume formula

*
, we acquire the following:
*
.

The radius 

*
defines the bounds together being
*
. Next, 
*
have the right to be found by understanding that the value is the street from the optimal to the bottom that the one at any kind of given point along
*
. The length of one side of the it is intended triangle, therefore, is
*
.

Putting it all together, the complying with is obtained:

*
 
*

*Note: the difficulty did not specify if the overcome sections to be perpendicular come the 

*
or
*
axis. Because the basic is a circle, this should not readjust the result volume. The only distinction should be the usage of
*
 or 
*
as variables in the exactly expression.


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Example concern #6 : uncover Cross Sections: triangles & Semicircles


Find the expression because that the volume the the solid who cross-sections are semicircles perpendicular to the 

*
axis and also whose base is bounded by 
*
and
*
.


Possible Answers:
Correct answer:

*


Explanation:

Since the cross-sections space perpendicular come the 

*
axis, the volume expression will certainly be in regards to
*

The area the a semicircle is

*
. By using this formula to our general volume formula
*
, we gain the following:
*
.

Since the region bounded through

*
and 
*
is the base of the solid, the intersection points of these attributes will create the bounds because that the volume expression. This points space
*
 and
*
. Because the expression is in terms of
*
, the 
*
collaborates can it is in referenced for the bounds.

Next, one expression for 

*
need to be determined. Since the radius 
*
is half the diameter that the semicircle, and also the diameter of the semicircle is the size stretching in between the functions 
*
and also
*
, the expression that the radius is the following:
*
. Simplified, this reads
*
.

Putting this every together, we find the following:

*


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Example inquiry #7 : find Cross Sections: triangles & Semicircles


Find the volume that the solid who cross-sections space semicircles and also whose base is bounded through the circle

*
.


Possible Answers:
Correct answer:

*


Explanation:

The base is characterized by the complying with formula:

*
. Therefore, the radius the the base is
*
. The radius 
*
specifies the bounds as being 
*

The exactly formula for the area the a semicircle is as follows:

*
, through r being the radius the the semicircle.

By using this formula to our basic volume formula

*
, we get the following:
*
.

Next, one expression for 

*
must be determined. The radius 
*
is fifty percent the diameter the the semicircle cross-section. The worth of 
*
is equivalent to the fifty percent the height of the base, or
*
. Therefore,
*

Putting this every together, we uncover the following:

*
 
*

*Note: the trouble did no specify if the cross sections were perpendicular to the 

*
or 
*
axis. Due to the fact that the basic is a circle, this need to not change the resulting volume. The only distinction should be the use of 
*
 or 
*
together variables in the correct expression.


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Example concern #8 : find Cross Sections: triangle & Semicircles


Identify the exactly expression because that the volume that the solid whose cross-sections space semicircles perpendicular to the 

*
axis and whose basic is bounded by 
*
and
*
.


Possible Answers:
Correct answer:

*


Explanation:

Since the cross-sections room perpendicular come the 

*
axis, the volume expression will certainly be in regards to
*

The area that a semicircle is

*
. By using this formula come our basic volume formula
*
, we gain the following:
*
.

Since the an ar bounded by 

*
and 
*
is the base of the solid, the intersection point out of these functions will create the bounds because that the volume expression. This points room
*
 and
*
. Due to the fact that the expression is in terms of
*
, the 
*
coordinates can it is in referenced for the bounds.

Next, an expression for 

*
must be determined. Since the radius 
*
is fifty percent the diameter the the semicircle, and also the diameter the the semicircle is the length stretching in between the functions 
*
and also
*
, the expression the the radius is the following:
*
.

Putting this every together, we find the following:

*


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Example concern #9 : discover Cross Sections: triangles & Semicircles


Identify the correct expression because that the volume of the solid whose cross-sections are semicircles perpendicular to the 

*
axis and also whose basic is bounded by 
*
and also
*
.


Possible Answers:
Correct answer:

*


Explanation:

Since the cross-sections are perpendicular come the 

*
axis, the volume expression will certainly be in terms of
*

The area the a semicircle is

*
. By applying this formula come our basic volume formula
*
, we gain the following:
*
.

Since the region bounded by 

*
 and 
*
is the basic of the solid, the intersection point out of these functions will develop the bounds because that the volume expression. This points are 
*
and
*
. Since the expression is in regards to
*
, the 
*
works with can be referenced for the bounds.

Next, an expression for 

*
should be determined. Due to the fact that the radius 
*
is fifty percent the diameter the the semicircle, and also the diameter the the semicircle is the size stretching in between the functions 
*
and also
*
, the expression of the radius is the following:
*
. This can be simplified: 
*

Putting this every together, we discover the following:

*


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Example question #10 : find Cross Sections: triangle & Semicircles


Identify the exactly expression because that the volume of the solid whose cross-sections space semicircles parallel come the y axis and whose base is bounded through

*
*
and also
*
.


Possible Answers:
Correct answer:

*


Explanation:

The cross-sections space parallel come the 

*
axis; this is another method of speak the cross-sections room perpendicular to the 
*
axis. Therefore, the volume expression will certainly be in regards to
*

The area the a semicircle is

*
. By applying this formula to our general volume formula
*
, we gain the following:
*
.

Since the region is bounded by

*
,
*
, and also
*
, the basic is the area in between the 
*
axis and 
*
top top the expression
*
. Since the expression is in regards to
*
, the interval 
*
will define the bounds.

See more: Thread: How Many Btus Does A Human Body Produce ? Was Morpheus Right

Next, an expression for 

*
need to be determined. Since the radius 
*
is fifty percent the diameter of the semicircle, and the diameter that the semicircle is the size stretching between 
*
and also the 
*
axis, the expression that the radius is the following:
*

Putting this all together, we uncover the following:

*


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