The region in the first quadrant bounded by y=6x^2 , 2x+y=8, and the y-axis is rotated about the line x=-1.
The volume of the resulting solid is:
To calculate the volume of the solid, we need to find the limits of integration and then set up an integral that represents the volume.
Step 1: Find the limits of integration
To determine the limits of integration, we need to find the intersection points of the curves.
First, let's find the intersection point between y = 6x^2 and 2x + y = 8.
Substitute y = 6x^2 into the second equation:
2x + 6x^2 = 8
Rearrange the equation to obtain a quadratic equation:
6x^2 + 2x - 8 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For this equation, a = 6, b = 2, and c = -8:
x = (-2 ± sqrt(2^2 - 4(6)(-8))) / (2(6))
x = (-2 ± sqrt(4 + 192)) / 12
x = (-2 ± sqrt(196)) / 12
x = (-2 ± 14) / 12
We have two possible values for x:
x1 = (-2 + 14) / 12 = 12 / 12 = 1
x2 = (-2 - 14) / 12 = -16 / 12 = -4/3
Now that we have the x-values, we can substitute them back into either equation to find the corresponding y-values.
For y = 6x^2:
y1 = 6(1)^2 = 6
For 2x + y = 8:
y2 = 8 - 2x2 = 8 - 2(-4/3) = 8 + 8/3 = 24/3 + 8/3 = 32/3
Therefore, the intersection points are (1, 6) and (-4/3, 32/3).
Step 2: Set up the integral
The region that needs to be rotated lies between the curves y = 6x^2 and 2x + y = 8. However, the rotation axis is x = -1.
To convert the rotation axis from x = -1 to the x-axis, we can shift the coordinate system by adding 1 to x for every point. This gives us the curves: y = 6(x + 1)^2 and 2(x + 1) + y = 8.
To calculate the volume using the disk method, we integrate from the smaller x-value (-4/3) to the larger x-value (1), using the formula:
V = π * ∫[a,b] (R^2 - r^2) dx
Where R represents the outer radius and r represents the inner radius.
The outer radius (R) is the distance from the rotation axis (x = -1) to the curve y = 6(x + 1)^2.
R = 1 - (-1) = 2
The inner radius (r) is the distance from the rotation axis (x = -1) to the line 2(x + 1) + y = 8, which is y - 8 = -2(x + 1).
To find r, substitute y = 6(x + 1)^2 into this equation:
6(x + 1)^2 - 8 = -2(x + 1)
6(x^2 + 2x + 1) - 8 = -2x - 2
6x^2 + 12x + 6 - 8 = -2x - 2
6x^2 + 14x - 4 = 0
3x^2 + 7x - 2 = 0
Using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For this equation, a = 3, b = 7, and c = -2:
x = (-7 ± sqrt(7^2 - 4(3)(-2))) / (2(3))
x = (-7 ± sqrt(49 + 24)) / 6
x = (-7 ± sqrt(73)) / 6
We have two possible values for x:
x3 = (-7 + sqrt(73)) / 6
x4 = (-7 - sqrt(73)) / 6
Substituting these values back into y = 6(x + 1)^2, we can find the corresponding y-values.
y3 = 6(x3 + 1)^2 = 6((-7 + sqrt(73)) / 6 + 1)^2
y4 = 6(x4 + 1)^2 = 6((-7 - sqrt(73)) / 6 + 1)^2
Now that we have the x and y-values, we can substitute them into the integral formula to find the volume:
V = π * ∫[a,b] (R^2 - r^2) dx
V = π * ∫[-4/3, 1] ((2^2) - ((6(x + 1)^2)^2)) dx
Simplifying the expression, we have:
V = π * ∫[-4/3, 1] (4 - 36(x + 1)^4) dx
Now, evaluate this integral to find the volume of the resulting solid.