Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x^2 and y = x about the line x = 7.
I have to use the volume by disks method, but I'm confused about how to set it up and evaluate
they intersect at (0,0) and when
.5x^2 = x
x^2 - 2x = 0
x(x-2) = 0
x = 0 or x = 2, then y = 0 or y = 2
at (0,0) and (2,2)
the disc will look like a washer,
outer radius = 7- y
inner radius = 7- √(2y)
volume = π∫( (7 - y)^2 - (7- √(2y))^2 ) dy from 0 to 2
expand (7 - y)^2 - (7- √(2y))^2 , integrate etc.
To find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x^2 and y = x about the line x = 7, we can use the volume by disks method.
Step 1: Sketch the region and the line of rotation. By drawing the given functions y = 0.5x^2 and y = x in the first quadrant, we can see that the region is a bounded region between these two curves. The line of rotation x = 7 is a vertical line passing through the point (7,0) in the xy-plane.
Step 2: Determine the limits of integration. To find the limits of integration, we need to identify the x-coordinates where the curves y = 0.5x^2 and y = x intersect. Setting the equations equal to each other:
0.5x^2 = x
Multiplying both sides by 2:
x^2 = 2x
Rearranging:
x^2 - 2x = 0
Factoring out an x:
x(x - 2) = 0
From this, we find that x = 0 and x = 2 are the x-coordinates where the curves intersect. Thus, the limits of integration for x will be from 0 to 2.
Step 3: Determine the radius of each disk. The radius of each disk is the distance from the line of rotation (x = 7) to each point on the curve y = 0.5x^2 or y = x. Since our line of rotation is a vertical line, the radius will be the difference between the x-coordinate of the line of rotation (7) and the x-coordinate of the curve.
For the curve y = 0.5x^2:
radius = 7 - x
For the curve y = x:
radius = 7 - x
Step 4: Determine the height (thickness) of each disk. The height of each disk is the differential change in x, dx. Since we are rotating the region in the x-direction, the height will be dx.
Step 5: Set up the integral. The volume of each disk is the area of the circular face, given by π * (radius)^2, multiplied by the thickness.
The integral to find the volume is:
V = ∫[from 0 to 2] π * [(7 - x)^2 - (7 - x)^2] dx
Simplifying, we can see that the second term inside the brackets cancels out, so we're left with:
V = ∫[from 0 to 2] π * (7 - x)^2 dx
Step 6: Evaluate the integral. Integrate the expression using the limits of integration.
V = π ∫[from 0 to 2] (49 -14x + x^2) dx
V = π [49x - 7x^2 + (1/3)x^3] evaluated from 0 to 2
V = π [(49(2) - 7(2)^2 + (1/3)(2)^3) - (49(0) - 7(0)^2 + (1/3)(0)^3)]
V = π [98 - 28 + (8/3) - 0]
V = π (42 + 8/3)
Finally, simplifying:
V = 42π + (8/3)π
Therefore, the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x^2 and y = x about the line x = 7 is given by the expression 42π + (8/3)π.
To find the volume of the solid obtained by rotating the region bounded by the curves y = 0.5x^2 and y = x about the line x = 7, we can use the method of cylindrical shells.
First, let's find the points of intersection between the two curves to determine the limits of integration. Setting the two equations equal to each other:
0.5x^2 = x
Rearranging the equation:
0.5x^2 - x = 0
Factoring out x:
x(0.5x - 1) = 0
So, we have two points of intersection: x = 0 and x = 2.
Now, let's set up the integral using the cylindrical shells method. The volume of a cylindrical shell can be calculated using the formula:
dV = 2πrhdx
where r is the radius of the shell, h is the height of the shell, dx is the thickness of the shell, and dV is the volume of the shell.
In this case, x ranges from 0 to 2, so the integral becomes:
V = ∫(from 0 to 2) 2π(7 - x)h dx
To express h in terms of x, we need to find the difference between the two curves at each x-value.
The equation for the first curve is y = 0.5x^2, and the equation for the second curve is y = x.
So, h = (0.5x^2) - (x) = 0.5x^2 - x
Substituting this expression for h into the integral, we have:
V = ∫(from 0 to 2) 2π(7 - x)(0.5x^2 - x) dx
Now, we can integrate this expression to find the volume:
V = 2π ∫(from 0 to 2) (7x^2 - x^3 - 14x + x^2) dx
V = 2π ∫(from 0 to 2) (8x^2 - x^3 - 14x) dx
To evaluate this integral, you can expand and simplify the expression. Then integrate each term one by one using the power rule of integration. Finally, evaluate the integral over the given limits from 0 to 2.
Once you evaluate the definite integral, the resulting value will give you the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x^2 and y = x about the line x = 7.