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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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.

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
.

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.

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

.

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:
.

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
.

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:
.

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

.

Explanation:

The base is characterized by the complying with formula:

. Therefore, the radius the the base is
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

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
.

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
.

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
.

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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