(a) Consider the region in the xy-plane consisting of the points (x; y) satisfying x > 0, y > 0, and lying between the curves y = x^2 + 1, y = 2x^2 2 and the two axes. Draw
a diagram of this region.
(b) This region is rotated about the y-axis to form a solid glass vase. Determine the
volume of glass in the vase.
y=x^2+1, y=2x^2-2 ***
Here is the sketch, the bottom diagram shows it best
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E2+%2B+1%2C+y+%3D+2x%5E2+-+2
We need their intersection:
2x^2 - 2 = x^2 + 1
x^2 = 3
x = ± √3 , y = 4
I will do it in two parts,
1. horizontal discs from y = -2 to y = 1, where I will use y = 2x^2 - 2
2x^2 = y+2
x^2 = y/2 + 1
Volume = π∫(y/2 + 1)dy/ from y = -2 to 1
= π[(1/4)y^2 + y] from -2 to 1
= π(1/4 + 1 - (4 - 4)
= 5π/4
2. now for the top part:
V = π∫(y/2 + 1 - (y - 1) dy/ from 1 to 4
= π∫( y/2 - y + 2) dy from 1 to 4
= π[y^2 /4 - y^2 /2 + 2y] from 1 to 4
= π( 4 - 8 + 8 - ( 1/4 - 1 + 2))
= π(4 - 5/4)
= 21π/4
total volume = 13π/2
check my arithmetic, should have written it out on paper first.
1. I get
= π[(1/4)y^2 + y] from -2 to 1
= π((1/4+1)-(1-2))
= 9π/4
2. I get 9π/4 also
making the total volume 9π/2
To draw the diagram of the region in the xy-plane, we need to identify the points that satisfy the given conditions: x > 0, y > 0, and lying between the curves y = x^2 + 1 and y = 2x^2 - 2, inclusive of the axes.
Step 1: Graph the curves y = x^2 + 1 and y = 2x^2 - 2.
To graph y = x^2 + 1, start by plotting a few points. For example, when x = -2, y = (-2)^2 + 1 = 5, and when x = 2, y = (2)^2 + 1 = 5. Connect the points in a smooth curve.
To graph y = 2x^2 - 2, similarly plot a few points. For example, when x = -2, y = 2(-2)^2 - 2 = 8, and when x = 2, y = 2(2)^2 - 2 = 6. Connect the points to form a smooth curve.
Step 2: Identify the points between the curves and within the specified quadrant.
The region we are interested in lies between the curves y = x^2 + 1 and y = 2x^2 - 2, within the first quadrant (x > 0, y > 0). Shade the region between these curves.
Step 3: Include the axes.
Lastly, we include the x and y axes to complete the diagram. Make sure to label the axes and any relevant points or values.
Now that we have the diagram, we can proceed to part (b) to determine the volume of glass in the vase.
To find the volume of the solid glass vase, we can use the method of cylindrical shells.
Step 1: Express the curves in terms of y.
Since we are rotating the region around the y-axis, we need to express the curves in terms of y instead of x.
For the curve y = x^2 + 1:
x = √(y - 1)
For the curve y = 2x^2 - 2:
x = √((y + 2)/2)
Step 2: Determine the height and radius of each cylindrical shell.
The height of each cylindrical shell will be the difference between the two x-values for a given y-value.
h = √(y - 1) - √((y + 2)/2)
The radius of each cylindrical shell will be the distance from the y-axis to the curve. Since we are rotating around the y-axis, the radius will simply be the x-value.
r = √(y - 1)
Step 3: Set up the integral for the volume of the vase.
The volume of the vase can be calculated as the integral over the range of y-values that define the region.
V = ∫ [a,b] (2πrh) dy
where a and b are the y-values where the curves intersect.
Step 4: Evaluate the integral.
Evaluate the integral to find the volume of the glass vase.
V = ∫ [a,b] (2π√(y - 1) * (√(y - 1) - √((y + 2)/2))) dy
After evaluating the integral, the result will give you the volume of the glass vase.