Question
Asked Oct 10, 2019

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer for y = x^3, y=1, x=2; about y=-3

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Expert Answer

Step 1

Given a solid obtained by rotating the region bounded by the following curves:

y x3,y 1,x2 and the region is about y = -3
To find: the volume of the solid obtained by rotating the region bounded by the
given curves about the specified line
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y x3,y 1,x2 and the region is about y = -3 To find: the volume of the solid obtained by rotating the region bounded by the given curves about the specified line

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Step 2

We begin with sketching the graph of the given curves.

 

y= r°
y=x°
2
2
y= 1
y= 1
-2
0
-2
0
x =2
x = 2
-2
y=3
y= -3
Fig. 1
Fig. 2
Fig. 1 represents the graph of the given curve and specified line. We see that the
region is bounded by the given function. The graphs intersect at y 1 and x 2,
so those are the bounds on our region
Fig. 2 represents rectangle of width Ax sketched in the region. We'll rotate the
rectangle about the line y = -3 and determine the volume of the washer and then
add them all up from x = 1 to x=2
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y= r° y=x° 2 2 y= 1 y= 1 -2 0 -2 0 x =2 x = 2 -2 y=3 y= -3 Fig. 1 Fig. 2 Fig. 1 represents the graph of the given curve and specified line. We see that the region is bounded by the given function. The graphs intersect at y 1 and x 2, so those are the bounds on our region Fig. 2 represents rectangle of width Ax sketched in the region. We'll rotate the rectangle about the line y = -3 and determine the volume of the washer and then add them all up from x = 1 to x=2

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Step 3

To define washer in th...

2
y-1
-2
Tout
A = 2
in
2
y= -3
Fig. 3
The washer is drawn to the right. In Fig. 3. The Volume Vw of the washer is
given by
Vw(area of the base)Ax
Where the base of the washer is the region formed by the two concentric circles.
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2 y-1 -2 Tout A = 2 in 2 y= -3 Fig. 3 The washer is drawn to the right. In Fig. 3. The Volume Vw of the washer is given by Vw(area of the base)Ax Where the base of the washer is the region formed by the two concentric circles.

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Tagged in

Math

Calculus

Integration