Lo.Lee.Ta.
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Hi, I'm still practicing how to find volume.
1. My problem is this:
"Find the volume of the solid described below:
The base of the solid is the disk x^2 + y^2 ≤ 4. The cross-sections by planes perpendicular to the y-axis between y=-2 and y=2 are isosceles right triangles with one leg in the disk."
2. I thought that since an isosceles right triangle is really just half of a square, I could appraoch this problem like every cross-section was a square and divide the volume in half at the end...
Using this method I have:
A = s^2
A= (2√(4-y^2))^2
A= 4(4-y^2)
A= 16 - 4y^2
This is the area if every cross-section was a square.V= 1/2 * 2 ∫0 to 2 of 16 - 4y^2 (The 1/2 on the outside is there because we need to find half of the volume to end up with isosceles right triangle cross-sections. The 2 is there because I changed the limits from -2 to 2 instead to 0 to 2.)
V= ∫0 to 2 of 16 - 4y^2
= 16y - 4*(y^3/3) |0 to 2
= 16(2) - 4*[(2)^3)/3]
= 32 - 10.67
= 21.33
But this is the wrong answer... Why is it wrong, though?
I avoided having to think about the cross-sections as triangles, because the area formula for that is:
1/2(b*h)
I am confused as to what to plug in for the b and h...
I know for an isoscleses right triangle, h = 1/2b
Would b = sqrt(-y^2 + 4)...?
And so Area= 1/2[sqrt(-y^2 + 4) * (1/2(sqrt(-y^2 + 4)))]
...I really don't know here. That's why I avoided triangles!
Please help!
Thank you VERY much! :)
1. My problem is this:
"Find the volume of the solid described below:
The base of the solid is the disk x^2 + y^2 ≤ 4. The cross-sections by planes perpendicular to the y-axis between y=-2 and y=2 are isosceles right triangles with one leg in the disk."
2. I thought that since an isosceles right triangle is really just half of a square, I could appraoch this problem like every cross-section was a square and divide the volume in half at the end...
Using this method I have:
A = s^2
A= (2√(4-y^2))^2
A= 4(4-y^2)
A= 16 - 4y^2
This is the area if every cross-section was a square.V= 1/2 * 2 ∫0 to 2 of 16 - 4y^2 (The 1/2 on the outside is there because we need to find half of the volume to end up with isosceles right triangle cross-sections. The 2 is there because I changed the limits from -2 to 2 instead to 0 to 2.)
V= ∫0 to 2 of 16 - 4y^2
= 16y - 4*(y^3/3) |0 to 2
= 16(2) - 4*[(2)^3)/3]
= 32 - 10.67
= 21.33
But this is the wrong answer... Why is it wrong, though?
I avoided having to think about the cross-sections as triangles, because the area formula for that is:
1/2(b*h)
I am confused as to what to plug in for the b and h...
I know for an isoscleses right triangle, h = 1/2b
Would b = sqrt(-y^2 + 4)...?
And so Area= 1/2[sqrt(-y^2 + 4) * (1/2(sqrt(-y^2 + 4)))]
...I really don't know here. That's why I avoided triangles!
Please help!
Thank you VERY much! :)