Question 1

Sketch the region bounded by the graphs of the given equations and find the area of their region.
$$x = y ^ { 2 } + 4 , x = y - 2 , y = - 2 , y = 2$$

Accepted Answer

The answer of Sketch the region bounded by the graphs...

Question 2

Find the volume of the solid obtained by rotating the region bounded by $y = 2 \sqrt [ 4 ] { x }$ and $y = 2 x$ about the line $y = 2$.

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 3

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the $y$-axis.
$$x ^ { 2 } - y ^ { 2 } = 16 , x \geq 0 , y = - 4 , y = 4$$

Accepted Answer

The answer of Find the volume of the solid generated...

Question 4

Sketch the region enclosed by  y=1+xy = 1 + \sqrt { x }y=1+x​  and  y=4+x4y = \frac { 4 + x } { 4 }y=44+x​ . Find the area of the region.

Accepted Answer

A)    323\frac { 32 } { 3 }332​ 
B)    423−4\frac { 4 \sqrt { 2 } } { 3 } - 4342c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​−4 
C)    2563256 \sqrt { 3 }2563c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)    3232\frac { 32 } { 3 } \sqrt { 2 }332​2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
E)    424 \sqrt { 2 }42c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​

Question 5

Find the area of the region bounded by the given curves.
 y=x2−6x,y=5x+12y = x ^ { 2 } - 6 x , y = 5 x + 12y=x2−6x,y=5x+12

Accepted Answer

A)   2197
B)    253\frac { 25 } { 3 }325​ 
C)    21973\frac { 2197 } { 3 }32197​ 
D)    21976\frac { 2197 } { 6 }62197​ 
E)   6

Question 6

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line.
$y = 1 - x ^ { 2 } , y = 0$; the line $y = 3$

Accepted Answer

The answer of Find the volume of the solid generated...

Question 7

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
$$y = 4 x , y = 4 x e ^ { 1 - x / 2 } , \text { about } y = 16$$

Accepted Answer

The answer of Use a computer algebra system to find...

Question 8

Find the volume of the solid obtained by rotating the region bounded by $y = x$ and $x = y ^ { 3 }$ about the $x$-axis.

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 9

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle.
 y=16−x2,y=−x+4;y = \sqrt { 16 - x ^ { 2 } } , y = - x + 4 ;y=16−x2​,y=−x+4;  the  yyy -axis

Accepted Answer

A)    643π\frac { 64 } { 3 } \pi364​π 
B)    128π\quad 128 \pi128π 
C)    643π\frac { 64 } { 3 } \pi364​π 
D)    128π128 \pi128π

Question 10

Find the volume of the solid obtained by rotating the region bounded by  y=xy = xy=x  and  x=y3x = y ^ { 3 }x=y3  about the  xxx -axis.
Select the correct answer.

Accepted Answer

A)    16π16 \pi16π 
B)    72\frac { 7 } { 2 }27​ 
C)    415π\frac { 4 } { 15 } \pi154​π 
D)    1835\frac { 18 } { 35 }3518​ 
E)    167π\frac { 16 } { 7 } \pi716​π

Question 11

$$\text { Find the volume of the solid that is obtained by revolving the region about the } x \text {-axis. }$$

Accepted Answer

The answer of \[\text { Find the volume of the...

Question 12

Sketch the region enclosed by the curves $y = x + 2 , y = 9 - x ^ { 2 } , x = - 2$, and $x = 2$. Find the area of the region correct to two decimal places.

Accepted Answer

The answer of Sketch the region enclosed by the curves...

Question 13

Sketch the region bounded by the graphs of the given equations and find the area of that region.
$$y = \sin 2 x , \mathrm { y } = 7 \cos x , x = \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 }$$

Accepted Answer

The answer of Sketch the region bounded by the graphs...

Question 14

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line.
 y=4−x2,y=0;y = 4 - x ^ { 2 } , y = 0 ;y=4−x2,y=0;  the line  y=6y = 6y=6

Accepted Answer

A)    70415π\frac { 704 } { 15 } \pi15704​π 
B)    140815π\frac { 1408 } { 15 } \pi151408​π 
C)    281615π\frac { 2816 } { 15 } \pi152816​π 
D)    14085π\frac { 1408 } { 5 } \pi51408​π

Question 15

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle.
$y = \sqrt { 16 - x ^ { 2 } } , y = - x + 4 ;$ the $y$-axis

Accepted Answer

The answer of Use the method of cylindrical shells to...

Question 16

Find the area of the shaded region.

Accepted Answer

A)   7
B)    712\frac { 7 } { 12 }127​ 
C)    16\frac { 1 } { 6 }61​ 
D)    76\frac { 7 } { 6 }67​

Question 17

Use the Midpoint Rule with  n=4n = 4n=4  to estimate the volume obtained by rotating about the region under the  yyy -axis the region under the curve.
 y=tan⁡x,0≤x≤π4y = 	an x , 0 \leq x \leq \frac { \pi } { 4 }y=tanx,0≤x≤4π​  The choices are rounded to the nearest hundredth. Select the correct answer.

Accepted Answer

A)    V=0.156V = 0.156V=0.156 
B)    V=0.491V = 0.491V=0.491 
C)    V=1.851V = 1.851V=1.851 
D)    V=0.825V = 0.825V=0.825 
E)    V=1.142V = 1.142V=1.142

Question 18

If  4 J4 \mathrm {~J}4 J  of work are needed to stretch a spring from  10 cm10 \mathrm {~cm}10 cm  to  12 cm12 \mathrm {~cm}12 cm  and another  20 J20 \mathrm {~J}20 J  are needed to stretch it from  12 cm12 \mathrm {~cm}12 cm  to  14 cm14 \mathrm {~cm}14 cm , what is the natural length of the spring? Round the answer to nearest integer.
Select the correct answer.

Accepted Answer

A)    12 cm12 \mathrm {~cm}12 cm 
B)    8 cm8 \mathrm {~cm}8 cm 
C)    13 cm13 \mathrm {~cm}13 cm 
D)    11 cm11 \mathrm {~cm}11 cm 
E)    10 cm10 \mathrm {~cm}10 cm

Question 19

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.
$x ^ { 2 } + ( y - 2 ) ^ { 2 } = 2 ^ { 2 }$ about the $y$-axis

Accepted Answer

The answer of The region bounded by the given curves...

Question 20

Find the area of the region bounded by the given curves.
 y=cos⁡x,y=sin⁡4x,x=0,x=π2y = \cos x , y = \sin 4 x , x = 0 , x = \frac { \pi } { 2 }y=cosx,y=sin4x,x=0,x=2π​

Accepted Answer

A)   4
B)   1
C)    13\frac { 1 } { 3 }31​ 
D)    14\frac { 1 } { 4 }41​ 
E)   2

Exam 5: Integrals

$\text { Find the volume of the solid that is obtained by revolving the region about the } x \text {-axis. }$ $\text { Find the volume of the solid that is obtained by revolving the region about the } x \text {-axis. }$

Correct Answer
verified

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 2 ^ { 2 }$ about the $y$ -axis

Correct Answer
verified

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. $y = \sqrt { 16 - x ^ { 2 } } , y = - x + 4 ;$ the $y$ -axis

Correct Answer
verified

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the $y$ -axis. $x ^ { 2 } - y ^ { 2 } = 16 , x \geq 0 , y = - 4 , y = 4$

Correct Answer
verified

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. $y = \sqrt { 16 - x ^ { 2 } } , y = - x + 4 ;$ the $y$ -axis

Correct Answer
verified

Sketch the region bounded by the graphs of the given equations and find the area of their region. $x = y ^ { 2 } + 4 , x = y - 2 , y = - 2 , y = 2$

Correct Answer
verified

Find the area of the region bounded by the given curves. $y = \cos x , y = \sin 4 x , x = 0 , x = \frac { \pi } { 2 }$

Correct Answer
verified

Sketch the region enclosed by $y = 1 + \sqrt { x }$ and $y = \frac { 4 + x } { 4 }$ . Find the area of the region.

Correct Answer
verified

Use the Midpoint Rule with $n = 4$ to estimate the volume obtained by rotating about the region under the $y$ -axis the region under the curve. $y = \tan x , 0 \leq x \leq \frac { \pi } { 4 }$ The choices are rounded to the nearest hundredth. Select the correct answer.

Correct Answer
verified

Correct Answer
verified

Find the area of the region bounded by the given curves. $y = x ^ { 2 } - 6 x , y = 5 x + 12$

Correct Answer
verified

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = 4 x , y = 4 x e ^ { 1 - x / 2 } , \text { about } y = 16$

Correct Answer
verified

Find the area of the shaded region. $Find the area of the shaded region. A) 7 B) \frac { 7 } { 12 } C) \frac { 1 } { 6 } D) \frac { 7 } { 6 }$

Correct Answer
verified

Sketch the region enclosed by the curves $y = x + 2 , y = 9 - x ^ { 2 } , x = - 2$ , and $x = 2$ . Find the area of the region correct to two decimal places.

Correct Answer
verified

Find the volume of the solid obtained by rotating the region bounded by $y = x$ and $x = y ^ { 3 }$ about the $x$ -axis.

Correct Answer
verified

Find the volume of the solid obtained by rotating the region bounded by $y = x$ and $x = y ^ { 3 }$ about the $x$ -axis. Select the correct answer.

Correct Answer
verified

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. $y = 1 - x ^ { 2 } , y = 0$ ; the line $y = 3$

Correct Answer
verified

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. $y = 4 - x ^ { 2 } , y = 0 ;$ the line $y = 6$

Correct Answer
verified

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y = \sin 2 x , \mathrm { y } = 7 \cos x , x = \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 }$

Correct Answer
verified

Find the volume of the solid obtained by rotating the region bounded by $y = 2 \sqrt [ 4 ] { x }$ and $y = 2 x$ about the line $y = 2$ .

Correct Answer
verified

Exam 5: Integrals

Find the volume of the solid that is obtained by revolving the region about the x-axis. \text { Find the volume of the solid that is obtained by revolving the region about the } x \text {-axis. } Find the volume of the solid that is obtained by revolving the region about the x-axis.

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The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x2+(y−2)2=22x ^ { 2 } + ( y - 2 ) ^ { 2 } = 2 ^ { 2 }x2+(y−2)2=22 about the yyy -axis

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the yyy -axis. x2−y2=16,x≥0,y=−4,y=4x ^ { 2 } - y ^ { 2 } = 16 , x \geq 0 , y = - 4 , y = 4x2−y2=16,x≥0,y=−4,y=4

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Sketch the region bounded by the graphs of the given equations and find the area of their region. x=y2+4,x=y−2,y=−2,y=2x = y ^ { 2 } + 4 , x = y - 2 , y = - 2 , y = 2x=y2+4,x=y−2,y=−2,y=2

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Find the area of the region bounded by the given curves. y=cos⁡x,y=sin⁡4x,x=0,x=π2y = \cos x , y = \sin 4 x , x = 0 , x = \frac { \pi } { 2 }y=cosx,y=sin4x,x=0,x=2π​

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Sketch the region enclosed by y=1+xy = 1 + \sqrt { x }y=1+x​ and y=4+x4y = \frac { 4 + x } { 4 }y=44+x​ . Find the area of the region.

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the area of the region bounded by the given curves. y=x2−6x,y=5x+12y = x ^ { 2 } - 6 x , y = 5 x + 12y=x2−6x,y=5x+12

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Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=4x,y=4xe1−x/2, about y=16y = 4 x , y = 4 x e ^ { 1 - x / 2 } , \text { about } y = 16y=4x,y=4xe1−x/2, about y=16

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Find the area of the shaded region.

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Sketch the region enclosed by the curves y=x+2,y=9−x2,x=−2y = x + 2 , y = 9 - x ^ { 2 } , x = - 2y=x+2,y=9−x2,x=−2 , and x=2x = 2x=2 . Find the area of the region correct to two decimal places.

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Find the volume of the solid obtained by rotating the region bounded by y=xy = xy=x and x=y3x = y ^ { 3 }x=y3 about the xxx -axis.

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Find the volume of the solid obtained by rotating the region bounded by y=xy = xy=x and x=y3x = y ^ { 3 }x=y3 about the xxx -axis. Select the correct answer.

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Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y=1−x2,y=0y = 1 - x ^ { 2 } , y = 0y=1−x2,y=0 ; the line y=3y = 3y=3

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Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y=4−x2,y=0;y = 4 - x ^ { 2 } , y = 0 ;y=4−x2,y=0; the line y=6y = 6y=6

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Sketch the region bounded by the graphs of the given equations and find the area of that region. y=sin⁡2x,y=7cos⁡x,x=π2,x=3π2y = \sin 2 x , \mathrm { y } = 7 \cos x , x = \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 }y=sin2x,y=7cosx,x=2π​,x=23π​

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Find the volume of the solid obtained by rotating the region bounded by y=2x4y = 2 \sqrt [ 4 ] { x }y=24x​ and y=2xy = 2 xy=2x about the line y=2y = 2y=2 .

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$\text { Find the volume of the solid that is obtained by revolving the region about the } x \text {-axis. }$ $\text { Find the volume of the solid that is obtained by revolving the region about the } x \text {-axis. }$

Correct Answer
verified

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 2 ^ { 2 }$ about the $y$ -axis

Correct Answer
verified

Correct Answer
verified

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the $y$ -axis. $x ^ { 2 } - y ^ { 2 } = 16 , x \geq 0 , y = - 4 , y = 4$

Correct Answer
verified

Correct Answer
verified

Sketch the region bounded by the graphs of the given equations and find the area of their region. $x = y ^ { 2 } + 4 , x = y - 2 , y = - 2 , y = 2$

Correct Answer
verified

Find the area of the region bounded by the given curves. $y = \cos x , y = \sin 4 x , x = 0 , x = \frac { \pi } { 2 }$

Correct Answer
verified

Sketch the region enclosed by $y = 1 + \sqrt { x }$ and $y = \frac { 4 + x } { 4 }$ . Find the area of the region.

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Find the area of the region bounded by the given curves. $y = x ^ { 2 } - 6 x , y = 5 x + 12$

Correct Answer
verified

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = 4 x , y = 4 x e ^ { 1 - x / 2 } , \text { about } y = 16$

Correct Answer
verified

Find the area of the shaded region. $Find the area of the shaded region. A) 7 B) \frac { 7 } { 12 } C) \frac { 1 } { 6 } D) \frac { 7 } { 6 }$

Correct Answer
verified

Sketch the region enclosed by the curves $y = x + 2 , y = 9 - x ^ { 2 } , x = - 2$ , and $x = 2$ . Find the area of the region correct to two decimal places.

Correct Answer
verified

Find the volume of the solid obtained by rotating the region bounded by $y = x$ and $x = y ^ { 3 }$ about the $x$ -axis.

Correct Answer
verified

Find the volume of the solid obtained by rotating the region bounded by $y = x$ and $x = y ^ { 3 }$ about the $x$ -axis. Select the correct answer.

Correct Answer
verified

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. $y = 1 - x ^ { 2 } , y = 0$ ; the line $y = 3$

Correct Answer
verified

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. $y = 4 - x ^ { 2 } , y = 0 ;$ the line $y = 6$

Correct Answer
verified

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y = \sin 2 x , \mathrm { y } = 7 \cos x , x = \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 }$

Correct Answer
verified

Find the volume of the solid obtained by rotating the region bounded by $y = 2 \sqrt [ 4 ] { x }$ and $y = 2 x$ about the line $y = 2$ .

Correct Answer
verified