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Unit 8: Applications of Integration
8.3 Finding the Volume of a Solid with Known Cross Sections
Setting up integrals to find volumes:
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Cards (52)
The area of the cross section must be a function of the variable along which the integration occurs.
True
The limits of integration, \( a \) and \( b \), are constants that define the interval along the x-axis.
True
Match the cross-section shape with its area formula:
Square ↔️ (side)^2
Rectangle ↔️ length × width
Triangle ↔️ 1/2 × base × height
Circle ↔️ \pi r^2
The volume of a solid can be found by integrating the
area
of its cross sections along a specified axis.
The area of the cross section at position \( x \) is denoted by \( A(x) \) in the
volume formula
.
True
The volume of the solid with square cross sections and side \( s(x) = x^2 \) is \( \frac{243}{5} \)
cubic
units.
True
The area of a circle is calculated using the formula
\pi r^2
The volume of the solid is found by integrating the area function from
0
to 3.
Common shapes for cross sections include squares, rectangles, triangles, and
semicircles
.
True
The general formula for finding the volume of a solid using known cross sections is
V = \int_{a}^{b} A(x) dx
Match the cross-section shape with its area formula:
Square ↔️ (side)^2
Rectangle ↔️ length × width
Triangle ↔️ 1/2 × base × height
Circle ↔️ \pi r^2
The definite integral to find the volume of a solid with square cross sections and side \( s(x) = x^2 \) is \( V = \int_{0}^{3}
x^4
In the example with square cross sections and side \( s(x) = x^2 \), the area function \( A(x) \) is equal to
x^4
The area formula for a triangle cross section is
1/2 × base × height
Common cross-section shapes include squares, rectangles, triangles, and
semicircles
What is the cross-sectional shape in the example solid mentioned?
Square
What is the volume of the solid in cubic units?
243
5
\frac{243}{5}
5
243
The area function \( A(x) \) represents the area of each cross section as a function of
x
.
What is the width function for the rectangular cross sections?
w
(
x
)
=
w(x) =
w
(
x
)
=
2
x
2x
2
x
What is the general formula for finding the volume of a solid using cross sections?
V
=
V =
V
=
∫
a
b
A
(
x
)
d
x
\int_{a}^{b} A(x) dx
∫
a
b
A
(
x
)
d
x
Steps to find the volume of a solid with known cross sections:
1️⃣ Identify the type of cross sections
2️⃣ Determine the area function \( A(x) \)
3️⃣ Set up the volume integral
4️⃣ Integrate and evaluate
The area function for square cross sections with side length \( s(x) = x^2 \) is
A
(
x
)
=
A(x) =
A
(
x
)
=
x
4
x^{4}
x
4
True
The area formula for a square is
(side)^2
The area formula for a triangle is
1
2
×
b
a
s
e
×
h
e
i
g
h
t
\frac{1}{2} \times base \times height
2
1
×
ba
se
×
h
e
i
g
h
t
True
What is the area function for square cross sections with side length
s
(
x
)
=
s(x) =
s
(
x
)
=
x
2
x^{2}
x
2
?
A
(
x
)
=
A(x) =
A
(
x
)
=
x
4
x^{4}
x
4
Consider a solid with rectangular cross sections where
l
(
x
)
=
l(x) =
l
(
x
)
=
x
x
x
and
w
(
x
)
=
w(x) =
w
(
x
)
=
2
x
2x
2
x
. The area function is A(x) = 2x^2</latex>
The limits of integration are determined by the bounds of the solid along the
x-axis
.
True
What is the antiderivative of
A
(
x
)
=
A(x) =
A
(
x
)
=
2
x
2
2x^{2}
2
x
2
?
F
(
x
)
=
F(x) =
F
(
x
)
=
2
3
x
3
\frac{2}{3}x^{3}
3
2
x
3
The volume of a solid with
A
(
x
)
=
A(x) =
A
(
x
)
=
2
x
2
2x^{2}
2
x
2
and limits
a
=
a =
a
=
1
1
1
and
b
=
b =
b
=
4
4
4
is
126
3
\frac{126}{3}
3
126
The limits of integration in the volume formula represent the boundaries along the
axis of integration
.
True
Steps to find the volume of a solid with square cross sections:
1️⃣ Identify the area function: \( A(x) = (x^2)^2 = x^4 \)
2️⃣ Set up the integral: \( V = \int_{0}^{3} x^4 dx \)
3️⃣ Integrate and evaluate: \( V = \left[ \frac{x^5}{5} \right]_0^3 = \frac{243}{5} \)
The variable \( V \) in the volume formula represents the
volume
The volume of the solid with square cross sections and side \( s(x) = x^2 \) is \( \frac{243}{5} \) cubic
units
.
True
Steps to find the volume of a solid with square cross sections:
1️⃣ Identify the area function: \( A(x) = (x^2)^2 = x^4 \)
2️⃣ Set up the integral: \( V = \int_{0}^{3} x^4 dx \)
3️⃣ Integrate and evaluate: \( V = \left[ \frac{x^5}{5} \right]_0^3 = \frac{243}{5} \)
Steps to find the volume of a solid with square cross sections:
1️⃣ Identify the area function: \( A(x) = (x^2)^2 = x^4 \)
2️⃣ Set up the integral: \( V = \int_{0}^{3} x^4 dx \)
3️⃣ Integrate and evaluate: \( V = \left[ \frac{x^5}{5} \right]_0^3 = \frac{243}{5} \)
What is the formula for the area of a triangle?
1
2
×
b
a
s
e
×
h
e
i
g
h
t
\frac{1}{2} \times base \times height
2
1
×
ba
se
×
h
e
i
g
h
t
The area function for the example solid is
A
(
x
)
=
A(x) =
A
(
x
)
=
x
4
x^{4}
x
4
True
The limits of integration in the volume formula represent the start and end points along the
x-axis
.
True
Knowing the cross section shape determines the area formula needed to find the
volume
.
What is the formula for the area function of rectangular cross sections?
A
(
x
)
=
A(x) =
A
(
x
)
=
l
(
x
)
×
w
(
x
)
l(x) \times w(x)
l
(
x
)
×
w
(
x
)
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