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Calculus AB : Find Cross-Sections: Squares & Rectangles

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Example Question #1 : Find Cross Sections: Squares & Rectangles

Find the volume of a pyramid whose base is a square with sides of length and whose height is .

Possible Answers:

Correct answer:

Explanation:

First, it is important to consider the shape of this solid. This solid is a pyramid, with one square face and four triangular faces. Through a relationship of similar triangles, we are able to relate the known information (a height of and a base side length of ) to our general variables for side length (x) and height of the pyramid (y) . We can think of this plotted on the coordinate plane, with the width of the pyramid solid being in the direction.

$\frac{x}{3}=\frac{y}{10} \Rightarrow x=\frac{3y}{10}$

Since the side length can be squared to find a general formula for the area of the pyramid’s base (A=x^2) , this can be applied to the volume using cross-sections formula $(V=\int_{a}^{b}A(x) dx)$ as the next step. Note that the equation $x=\frac{3y}{10}$ solved for above is in terms of . Our new volume function, therefore, is also in terms of $y: V=\int_{a}^{b}(\frac{3y}{10})^2 \: dy$

Because the pyramid reaches a maximum height of , and we assume the pyramid’s starting height is at y=0 , the appropriate bounds for the integral expression are: $0\leq y\leq 10$ .

To wrap up this problem, combine the above information into one cohesive expression:

$V=\int_{0}^{10}(\frac{3y}{10})^2 dy =\frac{9}{100} \int_{0}^{10}y^2dy =\frac{9}{100}[\frac{1}{3}y^3] |_0^{10}=\frac{9}{100}*\frac{1}{3}*1000=30$

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Example Question #2 : Find Cross Sections: Squares & Rectangles

Find the volume of the solid whose base is bounded by y=x+4 and y=x^2-2 and whose cross-sections are rectangles of height and perpendicular to the axis.

Possible Answers:

62.5

50.5

Correct answer:

62.5

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

The area of a rectangle is A=l*w . By applying this formula to our general volume formula $(V=\int_{a}^{b}A(x) dx)$ , we get the following: $V=\int_{a}^{b}lw \: dx$ .

Next, an expression for must be determined. The problem specifies the length (or height) of the rectangle cross-sections is . This just leaves the value of w to be found. The width of the rectangle will vary as the region creating the base of the solid varies. The region is defined by the area enclosed between y=x+4 and y=x^2-2 , therefore, w=(x+4)-(x^2-2)=6+x-x^2 .

Since the region is bounded by y=x+4 and y=x^2-2 , the base has the following domain: $-2\leq x\leq 3$ .

Putting this all together, we find the following:

$V=\int_{-2}^{3}(3)(6+x-x^2) dx$ $= 3[6x+\frac{1}{2}x^2-\frac{1}{3}x^3] |_{-2}^3=3(18+\frac{9}{2}-9+12-2-\frac{8}{3})=62.5$

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Example Question #3 : Find Cross Sections: Squares & Rectangles

Identify the correct expression for the volume of a solid whose base is bounded by y=x-3 , y=0 and y=5-x and whose cross-sections are squares perpendicular to the axis.

Possible Answers:

$V=\int_{3}^{5}4x^2-32x+64 \: dx$

$V=\int_{3}^{5}y^2 \: dy$

$V=\int_{3}^{5}4y^2-32y+64 \: dy$

$V=\int_{3}^{5}x^2 \: dx$

Correct answer:

$V=\int_{3}^{5}4x^2-32x+64 \: dx$

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

The area of a square is A=s^2 , where s is the side length of the square. By applying this formula to our general volume formula $(V=\int_{a}^{b}A(x) \: dx)$ , we get the following: $V=\int_{a}^{b}s^2 \: dx$ .

Next, an expression for s2 must be determined. Because s should be the width of the solid’s base, the expression for that length can be used to solve for s2.

s=(x-3)-(5-x)=2x-8

s^2=(2x-8)^2=4x^2-32x+64

Since the region is bounded by y=x-3 and y=5-x , the base has the following domain: 3x5 .

Putting this all together, we find the following:

$V=\int_{3}^{5}4x^2-32x+64 \: dx$

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Example Question #4 : Find Cross Sections: Squares & Rectangles

Identify the correct expression for the volume of a solid whose base is bounded by a disk of radius and whose cross-sections are squares parallel to the axis.

Possible Answers:

$4\int_{-R}^{R}y^2 \: dy$

$\int_{-R}^{R}R^2 \: dy$

$4\int_{-R}^{R}R^2-y^2 \: dy$

$\int_{-R}^{R}R^2-x^2 \: dx$

Correct answer:

$4\int_{-R}^{R}R^2-y^2 \: dy$

Explanation:

The cross-sections are parallel to the axis; in other words, they are perpendicular to the axis. This indicates that the expression should be in terms of .

Because the disk is of radius , the base is defined by the following formula: x^2+y^2=R^2 .

The area of a square is A=s^2 , with s being the side length. By applying this formula to our general volume formula $(V=\int_{a}^{b}A(y) \: dy)$ , we get the following: $V=\int_{a}^{b}s^2 \: dy$ .

The radius defines the bounds as being $-R\leq y\leq R$ . Next, can be found by understanding that differs as the width of the circle changes. The value of is the distance from one side of the circle to the other at any given point along $-R\leq y\leq R$ . The length of one side of the square, therefore, is $s=2\sqrt{R^2-y^2}$ .

Putting it all together, the following is obtained:

$V=\int_{-R}^{R}(2\sqrt{R2-y^2})^2\: dy= 4\int_{-R}^{R}R^2-y^2 dy$

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Example Question #5 : Find Cross Sections: Squares & Rectangles

Let be the region bounded by y=e^x , x=0 , and x=4 . Find the volume of the solid whose base is region and whose cross-sections are squares perpendicular to the axis.

Possible Answers:

$\frac{e^4-1}{2}$

e^8

e^4

$\frac{e^8-1}{2}$

Correct answer:

$\frac{e^8-1}{2}$

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

Next, an expression for s^2 must be determined. Because s should be the width of the solid’s base, the expression for that length can be used to solve for s^2 .

s=e^x-0

$s^2=e^{2x}$

Since the region is bounded by x=0 and x=4 , the base has the following domain: $0\leq x\leq 4$ .

Putting this all together, we find the following:

$V=\int_{0}^{4}e^{2x} \: dx =[\frac{1}{2}e^{2x}] |_0^4=\frac{1}{2}(e^8-e^0)=\frac{e^8-1}{2}$

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Example Question #6 : Find Cross Sections: Squares & Rectangles

Let be the region bounded by y=x^2-9 and y=0 . Find the volume of the solid whose base is region and whose cross-sections are rectangles perpendicular to the axis with height .

Possible Answers:

278

265

360

300

Correct answer:

360

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

The area of a rectangle is A=l*w . By applying this formula to our general volume formula $(V=\int_{a}^{b}A(x) \: dx)$ , we get the following: $V=\int_{a}^{b}lw \: dx$ .

Since the region is bounded by y=x^2-9 and y=0 , the base has the following domain: $-3\leq x\leq 3$ .

Putting this all together, we find the following:

$V=\int_{-3}^{3}10(9-x^2) dx =10[9x-\frac{1}{3}x^3] |_{-3}^3=10(27-9+27-9)=360$

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Example Question #7 : Find Cross Sections: Squares & Rectangles

Let be the region bounded by y=x^2+3 , x=0 and y=12 . Find the volume of the solid whose base is region and whose cross-sections are rectangles perpendicular to the axis with height .

Possible Answers:

108

Correct answer:

108

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

The area of a rectangle is A=l*w . By applying this formula to our general volume formula $(V=\int_{a}^{b}A(y) \: dy)$ , we get the following: $V=\int_{a}^{b}lw \: dy$ .

Next, an expression for must be determined. The problem specifies the length (or height) of the rectangle cross-sections is . This just leaves the value of to be found. The width of the rectangle will vary as the region creating the base of the solid varies. The region is defined by the area enclosed between y=x2+3 , x=0 and y=12 . The function y=x^2+3 is rewritten in terms of to become $x=\sqrt{y-3}$ , because the final expression should reflect the fact that the cross sections should be written in terms of . Therefore, $w=\sqrt{y-3} -0 =\sqrt{y-3 }$ .

Since the region is bounded by y=x^2+3 and y=12 , the base has the following domain: $3\leq y\leq 12$ .

Putting this all together, we find the following:

$V=\int_{3}^{12}6(\sqrt{y-3}) dy$ $=6\int_{3}^{12}(y-3)^{\frac{1}{2}} dy =6[\frac{2}{3}(y-3)^{\frac{3}{2}}] |_3^{12} =4(27-0)=108$

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Example Question #8 : Find Cross Sections: Squares & Rectangles

Identify the correct expression for the volume of a solid whose base is bounded by y=ln(x)+2 and the axis along $1\leq x\leq 4$ , and whose cross-sections are rectangles perpendicular to the axis with a height three times the width.

Possible Answers:

$3\int_{1}^{4}ln(x)^2+4ln(x)+4 \: dx$

$3\int_{1}^{4}ln(x)^2+4 \: dx$

$3\int_{1}^{4}ln(x)^2 \: dx$

$3\int_{1}^{4}ln(x)+2ln(x) \: dx$

Correct answer:

$3\int_{1}^{4}ln(x)^2+4ln(x)+4 \: dx$

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

The area of a rectangle is A=l*w . By applying this formula to our general volume formula $(V=\int_{a}^{b}A(x) \: dx)$ , we get the following: $V=\int_{a}^b{}lw \: dx$ .

Next, an expression for must be determined. The problem specifies the length (or height) of the rectangle cross-sections is three times the value of the width, or l=3w . The volume expression can now be modified: $V=\int_{a}^{b}3w^2 \: dx$

This just leaves the value of to be found. The width of the rectangle will vary as the region creating the base of the solid varies. The region is defined by the area enclosed between y=ln(x)+2 and y=0 , along $1\leq x\leq 4$ . Therefore, w=ln(x)+2-0 .

Putting this all together, we find the following:

$V=\int_{1}^{4}3(ln(x)+2)^2 dx=3\int_{1}^{4}ln(x)^2+4ln(x)+4 \: dx$

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Example Question #9 : Find Cross Sections: Squares & Rectangles

Identify the correct expression for the volume of a solid whose base is bounded by y=x^5-x^3+4 and y=4 and whose cross-sections are squares perpendicular to the axis.

Possible Answers:

$V=\int_{0}^{4}x^6-2x^8+x^{10} \: dx$

$V=\int_{0}^{1}x^6-2x^8+x^{10} \: dx$

$V=\int_{0}^{4}x^5-x^3-4x \: dx$

$V=\int_{0}^{1}x^3- x^5 \: dx$

Correct answer:

$V=\int_{0}^{1}x^6-2x^8+x^{10} \: dx$

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

The area of a square is A=s^2 , where is the side length of the square. By applying this formula to our general volume formula $(V=\int_{a}^{b}A(x) \: dx)$ , we get the following: $V=\int_{a}^{b}s^2\: dx$ .

Next, an expression for s^2 must be determined. Because s should be the width of the solid’s base, the expression for that length can be used to solve for s^2 .

s=(4)-(x^5-x^3+4)=x^3-x^5

$s^2=(x^3-x^5)^2=x^6-2x^8+x^{10}$

Since the region is bounded by y=x^5-x^3+4 and y=4 , the base has the following domain: $0\leq x\leq 1$ .

Putting this all together, we find the following:

$V=\int_{0}^{1}x^6-2x^8+x^{10} \: dx$

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Example Question #10 : Find Cross Sections: Squares & Rectangles

Identify the correct expression for the volume of a solid whose base is bounded by y=sin(x) and y=0 along and whose cross-sections are squares perpendicular to the axis.

Possible Answers:

$V=\int_{\frac{\pi}{2}}^{\pi}sin^2(x)\: dx$

$V=\int_{\frac{\pi}{2}}^{\pi}sin (x)\: dx$

$V=\int_{\frac{\pi}{2}}^{\pi}sin^2(y)\: dy$

$V=\int_{0}^{\pi}sin (x) - \frac{\pi}{2}\: dx$

Correct answer:

$V=\int_{\frac{\pi}{2}}^{\pi}sin^2(x)\: dx$

Explanation:

The cross-sections are perpendicular to the axis; therefore, the volume expression will be in terms of .

Next, an expression for s^2 must be determined. Because should be the width of the solid’s base, the expression for that length can be used to solve for s^2 .

s=sin(x)-0

s^2=sin^2(x)

Putting this all together, we find the following:

$V=\int_{\frac{\pi}{2}}^{\pi}sin^2(x) \: dx$

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Calculus AB : Find Cross-Sections: Squares & Rectangles

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Find Cross Sections: Squares & Rectangles

Example Question #2 : Find Cross Sections: Squares & Rectangles

Example Question #3 : Find Cross Sections: Squares & Rectangles

Example Question #4 : Find Cross Sections: Squares & Rectangles

Example Question #5 : Find Cross Sections: Squares & Rectangles

Example Question #6 : Find Cross Sections: Squares & Rectangles

Example Question #7 : Find Cross Sections: Squares & Rectangles

Example Question #8 : Find Cross Sections: Squares & Rectangles

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Example Question #10 : Find Cross Sections: Squares & Rectangles

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