Volume of Revolution

1

1141

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Can somebody help me?

1.) The region closed between the graphs of y = x and y = x[sup:3d43jdsq]2[/sup:3d43jdsq] is denoted by R. Find the volume generated when R is rotated through 360[sup:3d43jdsq]o[/sup:3d43jdsq] about

a.) the x-axis
b.) the y-axis


2.) A glass bowl is formed by rotating about the y-axis the region between the graphs of y = x[sup:3d43jdsq]2[/sup:3d43jdsq] and y = x[sup:3d43jdsq]3[/sup:3d43jdsq]. Find the volume of the glass in the bowl.

3.) The region enclosed by both axes, the line x = 2 and the curve y = 1/8x[sup:3d43jdsq]2[/sup:3d43jdsq] + 2 is rotated about the y-axis to form a solid. Find the volume of this solid.
 
1141 said:
Can somebody help me?

1.) The region closed between the graphs of y = x and y = x[sup:2tuzomkc]2[/sup:2tuzomkc] is denoted by R. Find the volume generated when R is rotated through 360[sup:2tuzomkc]o[/sup:2tuzomkc] about

a.) the x-axis
b.) the y-axis


2.) A glass bowl is formed by rotating about the y-axis the region between the graphs of y = x[sup:2tuzomkc]2[/sup:2tuzomkc] and y = x[sup:2tuzomkc]3[/sup:2tuzomkc]. Find the volume of the glass in the bowl.

3.) The region enclosed by both axes, the line x = 2 and the curve y = 1/8x[sup:2tuzomkc]2[/sup:2tuzomkc] + 2 is rotated about the y-axis to form a solid. Find the volume of this solid.
Click to expand...

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

Glenn and Kesav provided you detailed work for finding areas. Apply similar principle to calculate volume. Why is it that you cannot even start the problems?
 
I think all of these are correct:

5a.) a = 0, b = 1

f(x) = x
??[sub:2xzwntaj]a[/sub:2xzwntaj][sup:2xzwntaj]b[/sup:2xzwntaj] ?y[sup:2xzwntaj]2[/sup:2xzwntaj]dx
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?(x)[sup:2xzwntaj]2[/sup:2xzwntaj]dx
= [ ? 1/3x[sup:2xzwntaj]3[/sup:2xzwntaj]][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj]

? I(a) = ? 1/3(0)[sup:2xzwntaj]3[/sup:2xzwntaj]
= 0 ?

? I(b) = ? 1/3(1)[sup:2xzwntaj]3[/sup:2xzwntaj]
= 1/3 ?

f(x) = x[sup:2xzwntaj]2[/sup:2xzwntaj]
??[sub:2xzwntaj]a[/sub:2xzwntaj][sup:2xzwntaj]b[/sup:2xzwntaj] ?y[sup:2xzwntaj]2[/sup:2xzwntaj]dx
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?(x[sup:2xzwntaj]2[/sup:2xzwntaj])[sup:2xzwntaj]2[/sup:2xzwntaj]dx
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?x[sup:2xzwntaj]4[/sup:2xzwntaj]dx
= [ ? 1/5x[sup:2xzwntaj]5[/sup:2xzwntaj]][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj]

? I(a) = ? 1/5(0)[sup:2xzwntaj]5[/sup:2xzwntaj]
= 0 ?

? I(b) = ? 1/5(1)[sup:2xzwntaj]5[/sup:2xzwntaj]
= 1/5 ?

? 1/3? - 1/5?
= 2/15?

b.)

y = x[sup:2xzwntaj]2[/sup:2xzwntaj]
? x = -y[sup:2xzwntaj]1/2[/sup:2xzwntaj]

y = x
? x = y

c = 0, d = 1


f(y) = y
??[sub:2xzwntaj]c[/sub:2xzwntaj][sup:2xzwntaj]d[/sup:2xzwntaj] ?x[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?(y)[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= [ ? 1/3y[sup:2xzwntaj]3[/sup:2xzwntaj]][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj]

? I(c) = ? 1/3(0)[sup:2xzwntaj]3[/sup:2xzwntaj]
= 0 ?

? I(d) = ? 1/3(1)[sup:2xzwntaj]3[/sup:2xzwntaj]
= 1/3 ?

f(y) = -y[sup:2xzwntaj]1/2[/sup:2xzwntaj]
??[sub:2xzwntaj]c[/sub:2xzwntaj][sup:2xzwntaj]d[/sup:2xzwntaj] ?x[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?(-y[sup:2xzwntaj]1/2[/sup:2xzwntaj])[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?y dy
= [ ? 1/2y[sup:2xzwntaj]2[/sup:2xzwntaj]][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj]

? I(c) = ? 1/2(0)[sup:2xzwntaj]2[/sup:2xzwntaj]
= 0 ?

? I(d) = ? 1/2(1)[sup:2xzwntaj]2[/sup:2xzwntaj]
= 1/2 ?

? 1/2? - 1/3?
= 1/6?


8.)

y = x[sup:2xzwntaj]2[/sup:2xzwntaj]
? x = -y[sup:2xzwntaj]1/2[/sup:2xzwntaj]

y = x[sup:2xzwntaj]3[/sup:2xzwntaj]
? x = y[sup:2xzwntaj]1/3[/sup:2xzwntaj]

f(y) = -y[sup:2xzwntaj]1/2[/sup:2xzwntaj]
??[sub:2xzwntaj]c[/sub:2xzwntaj][sup:2xzwntaj]d[/sup:2xzwntaj] ?x[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?(-y[sup:2xzwntaj]1/2[/sup:2xzwntaj])[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?y dy
= [ ? 1/2y[sup:2xzwntaj]2[/sup:2xzwntaj]][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj]

? I(c) = ? 1/2(0)[sup:2xzwntaj]2[/sup:2xzwntaj]
= 0 ?

? I(d) = ? 1/2(1)[sup:2xzwntaj]2[/sup:2xzwntaj]
= 1/2 ?

f(y) = y[sup:2xzwntaj]1/3[/sup:2xzwntaj]
??[sub:2xzwntaj]c[/sub:2xzwntaj][sup:2xzwntaj]d[/sup:2xzwntaj] ?x[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?(y[sup:2xzwntaj]1/3[/sup:2xzwntaj])[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj] ?y[sup:2xzwntaj]2/3[/sup:2xzwntaj]dy
= [ ? 3/5y[sup:2xzwntaj]5/3[/sup:2xzwntaj]][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]1[/sup:2xzwntaj]

? I(c) = ? 3/5(0)[sup:2xzwntaj]5/3[/sup:2xzwntaj]
= 0 ?

? I(d) = ? 3/5(1)[sup:2xzwntaj]5/3[/sup:2xzwntaj]
= 3/5 ?

? 3/5? - 1/2?
= 1/0?


For 9.) I can't get it right. Can you tell me what I'm doing wrong?

x = 2

y = 1/8x[sup:2xzwntaj]2[/sup:2xzwntaj] + 2
? x[sup:2xzwntaj]2[/sup:2xzwntaj] = 8y - 16

c = 0, d = 2

f(y) = 2
??[sub:2xzwntaj]c[/sub:2xzwntaj][sup:2xzwntaj]d[/sup:2xzwntaj] ?x[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]2[/sup:2xzwntaj] ?(2)[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]2[/sup:2xzwntaj] ?4 dy
= [ ? 4y][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]2[/sup:2xzwntaj]

? I(c) = ? 4(0)
= 0 ?

? I(d) = ? 4(2)
= 8?

f(y) = 8y - 16
??[sub:2xzwntaj]c[/sub:2xzwntaj][sup:2xzwntaj]d[/sup:2xzwntaj] ?x[sup:2xzwntaj]2[/sup:2xzwntaj]dy
= ?[sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]2[/sup:2xzwntaj] ? 8y - 16 dy
= [ ? 4y[sup:2xzwntaj]2 - 16y[/sup:2xzwntaj]][sub:2xzwntaj]0[/sub:2xzwntaj][sup:2xzwntaj]2[/sup:2xzwntaj]

? I(c) = ? 4(0)[sup:2xzwntaj]2[/sup:2xzwntaj] - 16(0)
= 0 ?

? I(d) = ? 4(2)[sup:2xzwntaj]2[/sup:2xzwntaj] - 16(2)
= -24?
(The answer is supposed to be 9?)
 
\(\displaystyle 5(a) \ \pi\int_{0}^{1}(x^2-x^4)dx \ = \ \frac{2\pi}{15}, \ disc.\)

\(\displaystyle 5(b) \ \pi\int_{0}^{1}(y-y^2)dy \ = \ \frac{\pi}{6}, \ disc.\)

\(\displaystyle 8) \ \ \pi\int_{0}^{1}(y^{2/3}-y)dy \ = \ \frac{\pi}{10}, \ disc\)

\(\displaystyle 9) \ 2\pi\int_{0}^{2}x(x^2/8+2)dx \ = \ 9\pi, \ shell.\)
 
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