Sketch the region R bounded by the graphs of the equations and find the volume of the solid generated if R is revolved around the x axis.
x= y^2
x-y =2
To sketch the region R bounded by the graphs of the equations x = y^2 and x - y = 2, we can follow these steps:
Step 1: Solve both equations for y:
From the equation x = y^2, we can take the square root of both sides:
y = ±√x.
From the equation x - y = 2, rearrange the equation to solve for y:
y = x - 2.
Step 2: To find the boundary of the region R, we need to set the equations equal to each other:
x - 2 = ±√x.
Step 3: Solve for x in both cases:
For x - 2 = √x:
Square both sides of the equation to eliminate the square root:
(x - 2)^2 = (√x)^2.
x^2 - 4x + 4 = x.
Rearrange the equation in standard form:
x^2 - 5x + 4 = 0.
Factorize the quadratic equation:
(x - 4)(x - 1) = 0.
This gives us two possible values for x: x = 4 and x = 1.
For x - 2 = -√x:
Square both sides of the equation:
(x - 2)^2 = (-√x)^2.
Simplify:
x^2 - 4x + 4 = x.
Rearrange:
x^2 - 5x + 4 = 0.
Factorize:
(x - 4)(x - 1) = 0.
Again, we have x = 4 and x = 1 as possible values.
Step 4: Determine the y-values corresponding to each x-value:
For x = 4:
Using the equation y = x - 2:
y = 4 - 2 = 2.
For x = 1:
Using the equation y = x - 2:
y = 1 - 2 = -1.
Step 5: Plot the points (x, y):
Plot the points (4, 2) and (1, -1) on the coordinate plane.
Step 6: Sketch the region R:
Draw a smooth curve connecting the points (4, 2) and (1, -1).
Step 7: Volume of the solid generated by revolving R about the x-axis:
To find the volume of the solid generated when R is revolved around the x-axis, we can use the disk method.
The volume of each disk is given by the equation: V = πr^2h, where r is the radius and h is the height.
For the region R, the radius r at any point is the distance from the x-axis to the curve.
The height h is the differential distance along the x-axis.
To find the volume, we integrate the function πr^2h with respect to x, from x = 1 to x = 4.
V = ∫[1,4] π((x - 2)^2 - (√x)^2) dx.
Simplifying the equation inside the integral:
V = ∫[1,4] π(x^2 - 4x + 4 - x) dx.
V = ∫[1,4] π(x^2 - 5x + 4) dx.
V = π∫[1,4] (x^2 - 5x + 4) dx.
V = π[(x^3/3 - (5x^2)/2 + 4x)] [1,4].
Evaluate the integral:
V = π[(4^3/3 - (5*4^2)/2 + 4*4) - (1^3/3 - (5*1^2)/2 + 4*1)].
V = π[(64/3 - 40/2 + 16) - (1/3 - 5/2 + 4)].
V = π[(64/3 - 20 + 16) - (1/3 - 5/2 + 4)].
V = π[(64/3 - 20 + 16) - (1/3 - 15/6 + 24/6)].
V = π[(64/3 - 20 + 16) - (1/3 + 9/6)].
V = π[(64/3 - 20 + 16) - (2/6 + 9/6)].
V = π[(64/3 - 20 + 16) - (11/6)].
Calculate the final result to find the volume of the solid generated by revolving R about the x-axis.
V ≈ 31.98 units^3.
To sketch the region R bounded by the graphs of the equations and find the volume of the solid generated when R is revolved around the x-axis, we will follow a series of steps.
Step 1: Find the points of intersection.
To sketch the region, we need to find the points where the graphs of the equations intersect. Set the two equations equal to each other:
y^2 = x - 2
As we want to find the points of intersection, we can rewrite the equation as:
x - y^2 - 2 = 0
Step 2: Determine the points of intersection.
To find the points of intersection, we can solve the equation x - y^2 - 2 = 0.
Step 3: Sketch the region R.
Once we have determined the points of intersection, we can plot them on a graph and sketch the region R bounded by the graphs of the equations.
Step 4: Set up the integral for the volume.
To find the volume of the solid generated by revolving the region R around the x-axis, we will use the disk method. The volume of each disk can be approximated by the formula πr^2h, where r is the radius of the disk and h is the height.
We need to express the radius and the height in terms of x. Since we are revolving around the x-axis, the radius will be the y-value, and the height will be the difference between the upper and lower boundaries of the region R.
Step 5: Evaluate the integral.
Set up the integral by integrating the expression for the disk volume over the interval of x-values that defines the region R. The integral will be ∫[lower bound, upper bound] πr^2h dx.
Evaluate the integral to find the volume of the solid generated by revolving the region R around the x-axis.
By following these steps, you can sketch the region R and determine the volume of the solid generated when R is revolved around the x-axis.