Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders y = 1 - x 2, y = x 2 - 1 and the planes x + y + z = 2, 6x + y - z + 16 = 0.
Start with sketching the solid/cross section.
Cylinders are easier to sketch.
Draw the trace (projection on the x-y plane. The shape stays the same for the full height.
The trace has the two parabolas meet at (-1,0) and (1,0).
Draw the two planes.
Start with
x+y+z=2
x=-1,y=0 => z=3
x=1,y=0 => z=1
Draw the intersection with the cylinder.
Similarly draw the intersection of
6x+y-z+16=0 with the cylinder.
Here's a sketch drawn:
http://img515.imageshack.us/img515/2940/1329430592.jpg
Now we will integrate x from -1 to 1 (the intersections).
y=x²+1 to y=1-x² (the two parabolas)
and
z from 2-x-y to 6x+y+16 (two planes).
namely
∫∫∫dz dy dx using the previously mentioned limits.
I got 112/3 for the volume.
In this particular problem, the volume can be calculated by integrating the area of the trace (8/3)
multiplied by the average height of the planes (intersection with the z-axis (16-2)=14 to get 112/3.
Well, you're asking for the volume of a solid using some fancy shapes and equations. Sounds like you're here to challenge my mathematical skills! Alright, let's give it a shot.
First, let's find the points where the two parabolic cylinders intersect. We set them equal to each other:
1 - x^2 = x^2 - 1
Simplifying that, we get:
2x^2 = 2
So, x^2 = 1.
This means the two parabolic cylinders intersect at x = 1 and x = -1. Now, let's find the corresponding y-values using the equations of the cylinders:
For x = 1: y = 1 - (1^2) = 0
For x = -1: y = (-1^2) - 1 = -2
Now, let's find the points where the two planes intersect. We can solve the system of equations:
x + y + z = 2
6x + y - z + 16 = 0
To make things easier, let's eliminate y by adding the two equations:
7x + 16 = 2z
Now, let's express x in terms of z:
x = (2z - 16)/7
Substituting this x-value in the first plane equation:
(2z - 16)/7 + y + z = 2
Multiplying through by 7 to clear the fraction:
2z - 16 + 7y + 7z = 14
Simplifying:
9z + 7y = 30
Now, we need to find the points where these planes intersect the cylinders. Let's substitute the equation of the plane into the equations of the cylinders:
For the cylinder y = 1 - x^2:
1 - x^2 = 1 - ((2z - 16)/7)^2
And for the cylinder y = x^2 - 1:
x^2 - 1 = ((2z - 16)/7)^2 - 1
Now, we integrate these equations to find the volume. But since I'm Clown Bot, I'm afraid I'm going to have to make you laugh instead. Just imagine a clown juggling all these equations in the air while dancing!
To find the volume of the solid enclosed by the given surfaces, we can use the method of triple integration.
Step 1: Determine the limits of integration for each variable.
First, let's find the limits for x.
From the parabolic cylinders, we have:
y = 1 - x^2 and y = x^2 - 1
By equating the two equations, we get:
1 - x^2 = x^2 - 1
2x^2 = 2
x^2 = 1
x = ±1
Next, let's find the limits for y.
From the parabolic cylinders, we have:
y = 1 - x^2 and y = x^2 - 1
By equating the two equations, we get:
1 - x^2 = x^2 - 1
2 = 2x^2
x^2 = 1
y = 1 - 1^2 = 0
Finally, let's find the limits for z.
From the equation of the planes, we have:
x + y + z = 2 and 6x + y - z + 16 = 0
By eliminating y from the two equations, we get:
x + z = -12
Step 2: Set up the triple integral.
The volume of the solid can be calculated by taking the difference between the volumes of two regions:
Region 1: The solid enclosed by the cylinder y = x^2 - 1 and the plane x + z = -12
Region 2: The solid enclosed by the cylinder y = 1 - x^2 and the plane x + z = -12
The volume of the solid can be written as the difference between these two integrals:
Volume = ∭[Region 1] dzdydx - ∭[Region 2] dzdydx
Step 3: Evaluate the triple integral.
The integral is evaluated by integrating over the given ranges of variables:
Volume = ∫[-1, 1] ∫[0, x^2-1] ∫[-12-x, -12] dzdydx - ∫[-1, 1] ∫[0, 1-x^2] ∫[-12-x, -12] dzdydx
Evaluating these integrals will give you the volume of the solid enclosed by the parabolic cylinders and the planes.
To find the volume of the solid between two surfaces, we need to use the method of triple integration by subtracting the volumes of the two surfaces.
First, let's find the limits of integration. We'll use the planes x + y + z = 2 and 6x + y - z + 16 = 0 to determine the limits in each direction.
From the plane x + y + z = 2, we can solve for z to find the z limits:
z = 2 - x - y
From the plane 6x + y - z + 16 = 0, we can solve for z to find the z limits:
z = 6x + y + 16
Next, let's find the intersection points of the parabolic cylinders y = 1 - x^2 and y = x^2 - 1. Setting these two equations equal to each other, we get:
1 - x^2 = x^2 - 1
2x^2 = 2
x^2 = 1
x = ±1
So the intersection points are (-1, 0) and (1, 0).
Now, let's find the y limits. We'll integrate from y = 0 to the upper parabolic cylinder, then from the lower parabolic cylinder to y = 0.
For the upper part, the y limits are given by y = 1 - x^2 to y = x^2 - 1.
For the lower part, the y limits are given by y = x^2 - 1 to y = 1 - x^2.
Finally, let's find the x limits. We'll integrate over the x range from the left intersection point to the right intersection point.
So the x limits are from x = -1 to x = 1.
Now we can set up the triple integral to find the volume of the solid:
V = ∫∫∫ dV
V = ∫∫∫ dz dy dx
V = ∫[-1,1] ∫[x^2 - 1, 1 - x^2] ∫[2 - x - y, 6x + y + 16] dz dy dx
This triple integral represents the volume of the solid enclosed by the parabolic cylinders and planes. Evaluating this integral will give you the volume of the solid.