2. Let R be the region in the first quadrant bounded by the graphs of (x^2/9)+(y^2/81)=1 and 3x+y=9 .
a. Set up but do not evaluate an integral representing the area of R. Express the integrand as a function of a single variable.
b. Set up but do not evaluate an integral representing the volume of the solid generated when R is rotated about the x-axis. Express the integrand as a function of a single variable.
c. Set up but do not evaluate an integral representing the volume of the solid generated when R is rotated about the y-axis. Express the integrand as a function of a single variable.
First, see where the curves intersect. Just substitute y=9-3x into the equation for the ellipse to find they intersect at (0,9) and (3,0)
So, the integrals will all be for x in [0,33 or y in [0,9]
(a) ∫[0,3] y1-y2 dx
= ∫(9-3x) - sqrt(9-3x^2) dx
(b) volume using discs is
∫pi*(R^2-r^2) dx
where R = 9-3x and r=sqrt(9-3x^2)
= pi∫(9-3x)^2 - (9-3x^2) dx
(c) same idea, different axis
pi∫(3 - y/3)^2 - (3 - y^2/3) dy
a. To set up the integral representing the area of region R, we need to find the intersection points of the two curves.
First, let's rewrite the equation of the ellipse in standard form:
(x^2/9) + (y^2/81) = 1
Multiply both sides by 81 to eliminate the denominators:
9x^2 + y^2 = 81
Now we can substitute the equation of the line into the equation of the ellipse to find the intersection points:
9x^2 + (9 - 3x)^2 = 81
81x^2 - 54x + 81 = 81
81x^2 - 54x = 0
x(81x - 54) = 0
From this equation, we find two possible x-values: x = 0 and x = 2/3.
To determine the y-values corresponding to these x-values, we substitute them into the equation of the line:
For x = 0:
3(0) + y = 9
y = 9
For x = 2/3:
3(2/3) + y = 9
2 + y = 9
y = 7
Therefore, the two intersection points are (0, 9) and (2/3, 7). These will be the bounds of our integral.
Now, we need to express the integrand as a function of a single variable, which in this case will be y. We can rearrange the equation of the ellipse to solve for y:
9x^2 + y^2 = 81
y^2 = 81 - 9x^2
y = √(81 - 9x^2)
The integrand representing the area of region R is the length of an infinitesimally small line segment in the y-direction, which is given by √(81 - 9x^2). Therefore, the integral representing the area of R is:
∫[from 9 to 7] √(81 - 9x^2) dy
b. To set up the integral representing the volume of the solid generated when R is rotated about the x-axis, we need to use the method of disks or washers.
The equation of the ellipse is already in standard form, so we can express the radius in terms of y as:
r = x = √(9 - (9/81)y^2) = √(9 - (1/9)y^2)
The height of each disk or washer is given by the difference in y-values between the curve defined by the ellipse and the line, which is 9 - y.
Therefore, the integrand representing the volume of the solid generated when R is rotated about the x-axis is:
π * [√(9 - (1/9)y^2)]^2 * (9 - y) dy
c. To set up the integral representing the volume of the solid generated when R is rotated about the y-axis, we need to use the method of cylindrical shells.
The equation of the ellipse is already in standard form, so we can express the radius in terms of x as:
r = y = √(81 - 9x^2)
The height of each cylindrical shell is given by the difference in x-values between the curve defined by the ellipse and the line, which is (1/3)(9 - x).
Therefore, the integrand representing the volume of the solid generated when R is rotated about the y-axis is:
2π * x * √(81 - 9x^2) * (1/3)(9 - x) dx
To find the area of the region R, we need to determine the limits of integration and the integrand. The first step is to sketch the graphs and visualize the region R in the first quadrant.
1. Graphing the equations:
a) (x^2/9) + (y^2/81) = 1 represents an ellipse centered at the origin with major axis along the x-axis and minor axis along the y-axis. The equation of the ellipse can be rewritten as (x/3)^2 + (y/9)^2 = 1. The semi-major axis is 3, and the semi-minor axis is 9.
b) 3x + y = 9 represents a straight line passing through the points (0, 9) and (3, 0). To find the intersection points of the line and ellipse, we can substitute y = 9 - 3x into the equation of the ellipse and solve for x:
(x/3)^2 + ((9 - 3x)/9)^2 = 1
x^2/9 + (9 - 3x)^2/81 = 1
9x^2 + (9 - 3x)^2 = 81
Simplifying the equation will give us the values of x where the line and the ellipse intersect.
2. Integrating to find the area:
To set up the integral representing the area, we need to determine the limits of integration. Looking at the graph, we can see that the region R is bounded by the x-axis and the line 3x + y = 9.
a) Limits of integration:
The x-values where the line intersects the ellipse give us the limits of integration. We need to find these intersection points.
3. Solving for x:
9x^2 + (9 - 3x)^2 = 81
Expanding and simplifying the equation will give us a quadratic equation, which can be solved using factorizing or using the quadratic formula.
Once you solve for x and find the intersection points, these will give you the limits of integration for the area integral.
b) Integrand:
The integrand is the function representing the infinitesimal area element. In this case, we can express it as f(x) = y. To find the value of y in terms of x, we can substitute y = 9 - 3x into the equation of the ellipse: (x/3)^2 + ((9 - 3x)/9)^2 = 1.
By solving for y, we can express the integrand as a function of x.
3. Integrating to find the volume:
To find the volume of the solid generated when R is rotated about the x-axis or y-axis, we need to set up an appropriate integral.
a) Volume when R is rotated about the x-axis:
We create infinitesimally thin disks perpendicular to the x-axis. The volume of each disk can be expressed as π(r(x))^2 * dx, where r(x) is the distance from the axis of rotation (x-axis) to the ellipse at each x-value.
We need to express (r(x))^2 in terms of x.
b) Volume when R is rotated about the y-axis:
Similarly, when R is rotated about the y-axis, we create infinitesimally thin disks perpendicular to the y-axis. The volume of each disk can be expressed as π(r(y))^2 * dx, where r(y) is the distance from the axis of rotation (y-axis) to the ellipse at each y-value.
We need to express (r(y))^2 in terms of y.
By setting up the appropriate integrals with the correct limits of integration and integrands, we can find the volume when R is rotated about the x-axis and the y-axis.