# calculus

posted by .

The area bounded by the curve 2y^2=x and the line 4y=x is rotated around the y-axis. The volume of the resulting structure can be expressed as V=a/bπ, where a and b are coprime positive integers. What is the value of a+b?

• calculus -

something's wrong. The region is not closed.

• calculus -

i believe its bounded from y=0 to y=2

• calculus -

Ah. In that case, using discs (washers),

v = ∫[0,2] π(R^2-r^2) dy
where R = 4y and r = 2y^2, so

v = ∫[0,2] π((4y)^2-(2y^2)^2) dy
= 4π∫[0,2] 4y^2 - y^4 dy
= 1024/15 π

• calculus -

no, i think that
∫[0,2] π((4y)^2-(2y^2)^2) dy is 256/15π

• calculus -

you may be right. maybe I mixed in an unneeded factor of 4 somewhere.

## Similar Questions

1. ### calculus

The region bounded by the curve y=1รท(1+2x) , the line X=2 , the x-axis and the y axis is rotated completely about the y-axis. Show that the volume generated is 1/2π(4-ln5)
2. ### Calculus

1. Find the area of the region bounded by the curves and lines y=e^x sin e^x, x=0, y=0, and the curve's first positive intersection with the x-axis. 2. The area under the curve of y=1/x from x=a to x=5 is approximately 0.916 where …
3. ### calculus

Let V be the volume of the 3-dimensional structure bounded by the paraboloid z=1−x^2−y^2, planes x=0, y=0 and z=0 and by the cylinder x^2+y^2−x=0. If V=aπ/b, where a and b are coprime positive integers, what …
4. ### Calculus

The area bounded by the curve y = 2x^2-x^3 and line y=0 is rotated around the y-axis. The volume of the resulting structure can be expressed as V = a(pi)/b, where a and b are coprime positive integers. What is the value of a + b?
5. ### Calculus

Let V be the volume of the three-dimensional structure bounded by the region 0≤z≤1−x^2−y^2. If V=a/bπ, where a and b are positive coprime integers, what is a+b?
6. ### math calculus

The volume of the solid obtained by rotating the region bounded by y=x^2−2x and y=x about the line y=9, has the form a/bπ, where a and b are positive coprime integers. What is the value of a+b?
7. ### calculus

The area bounded by the curve 2y^2=x and the line 4y=x is rotated around the y-axis. The volume of the resulting structure can be expressed as V=a/bπ, where a and b are coprime positive integers. What is the value of a+b?
8. ### calculus

A 3-dimensional structure is obtained from rotating the parabola y=x^2 about the y-axis. Each second, 2π units^3 of water is being poured into the structure from the top. When 8π units^3 of water has been poured in the structure, …
9. ### calculus

Let V be the volume of the three-dimensional structure bounded by the region 0≤z≤1−x^2−y^2. If V=a/bπ, where a and b are positive coprime integers, what is a+b?
10. ### Calculus

The region in the first quadrant bounded by the x-axis, the line x = ln(π), and the curve y = sin(ex) is rotated about the x-axis. What is the volume of the generated solid?

More Similar Questions