disc and shells problems

[mr.burger]

1) The area bounded by y= e^(x) , y=0 , x=1 and x=2 is rotated around the x-axis. Find its volume. (you must use disks)


2) The area bounded by y=x^(3) , y=x^(2) +4 and x=0 is rotated about the y-axis. Find its volume. (you may use shells or disks)

 

[Gene]

dv = pi*e^(x)^2dx =
pi*e^(2x)dx
v = pi*Integ_{1 to 2}e^(2x)dx =
pi/2*e^(2x)]_{1 to 2} =
pi/2*(e^4-e^2)

By shells:
dv=2*pi*x*(x^2+4-x^3)dx =
2*pi*(x^3+4x-x^4)dx
v = 2*pi*Integ_{0 to 2}(x^3+4x-x^4)dx =
2*pi*((x^4)/4+2(x^2)-(x^5/5))]_{0 to 2} =
2*pi*((2^4)/4 + 2*(2^2)-(2^5)/5)

 

[soroban]

Hello, mr.burger!

These problems are pretty straight-forward.
Are you familiar with the Volume-of-Revolutions formulas!


Disks: . V . = . (pi) ∫_{a to b} (radius)² dx

Shells: . V . = . 2(pi) ∫_{a to b} (radius)(height) dx

1) The area bounded by y= e^x , y=0 , x=1 and x=2 is rotated around the x-axis.

Find its volume. (you must use disks)

. . . V . = . (pi) ∫_{1 to 2} (e^x)² dx

2) The area bounded by y=x³ , y=x² + 4 and x = 0 is rotated about the y-axis.

Find its volume. (you may use shells or disks)

The two curves intersect at (2,8).
The "radius" is x.
The "height" is: .(x² + 4) - x³

. . . V . = . 2(pi) ∫_{0 to 2} x(x² + 4 - x³) dx


The set-up is 90% of the work . . . the rest is "only" integrating and evaluating.

 

[mr.burger]

do you get 23.6 pi for the first and 5.6 pi for the second?

 

[mr.burger]

do you get 23.6 pi for the first and 11.2 pi for the second?

 

[soroban]

Hello, mr.burger!

Oh-oh . . . I got different answers . . .

1) The area bounded by y= e^x , y=0 , x=1 and x=2 is rotated around the x-axis.

Find its volume. (you must use disks)

V . = . (pi) ∫(e^x)² dx . [1 to 2]

V . = . (pi) ∫ e^2x dx . = . (pi/2) e^2x . [1 to 2]

V . = . (pi/2)(e⁴ - e²) .≈ .74.16

2) The area bounded by y=x³ , y=x² + 4 and x = 0 is rotated about the y-axis.

Find its volume. (you may use shells or disks)

V . = . 2(pi) ∫ x(x² + 4 - x³) dx . [0 to 2]

V . = . 2(pi)∫ (4x + x³ - x⁴) dx . = . 2(pi)[2x² + x⁴/4 - x⁵/5] .[0 to 2]

V . = . 2(pi) [(2·2² + 2⁴/4 - 2⁵/5) - (2·0² + 0⁴/4 - 0⁵/5)]

V . = . 2(pi)(68/5) . ≈ . 85.45

(And, as always, please check my work.)

 

[Gene]

Burger, yes your answers are correct.
Soroban,
We agree on the first. We agree as to the equation on the second but I get 35.19 60-32 = 28, not 68.
I have a hard time because WebTV doesn't like the codes you use. Integral, sqrt, theta... all come out as ?

 

[tkhunny]

First, agreed, ½*pi*e^2*(e^2 - 1) apprx 74.156
Second, (56/5)*pi apprx 35.186