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.
To find the set up for the integrals representing the area and volumes in this problem, we first need to understand the region R in the first quadrant bounded by the given curves.
a. Area of R:
To find the area of the region R, we need to find the points where the two curves intersect. The two curves are:
1) (x^2/9) + (y^2/81) = 1: This is an ellipse centered at the origin (0,0), with semi-major axis 3 and semi-minor axis 9.
2) 3x + y = 9: This is a straight line with slope -3 and intercepts the y-axis at 9.
To find the points of intersection, we need to solve the system of equations formed by the two curves.
First, isolate y in the equation of the line: y = 9 - 3x.
Then substitute this expression for y in the equation of the ellipse: (x^2/9) + ((9 - 3x)^2/81) = 1.
Now, solve this equation for x:
(x^2/9) + (81 - 54x + 9x^2)/81 = 1.
Multiplying through by 81 to clear the fractions, we get:
9x^2 + 81 - 54x + 9x^2 = 81.
18x^2 - 54x = 0.
Factoring out 18x, we get:
18x(x - 3) = 0.
So, we have two possible values for x: x = 0 and x = 3.
Plugging these values back into the equation of the line, we get the corresponding y-coordinates:
For x = 0, y = 9 - 3(0) = 9.
For x = 3, y = 9 - 3(3) = 9 - 9 = 0.
Therefore, the points of intersection are (0, 9) and (3, 0).
To set up the integral representing the area of R, we need to integrate the difference of the top and bottom functions with respect to x from x = 0 to x = 3.
The top function is given by the curve of the ellipse: y = √[81 - 81(x^2/9)] = 9√(1 - x^2/9).
The bottom function is given by the line: y = 9 - 3x.
So, the integral representing the area of R is:
∫[0 to 3] [9√(1 - x^2/9) - (9 - 3x)] dx.
b. Volume of the Solid Generated when R is Rotated about the x-axis:
To find the volume of the solid generated when region R is rotated about the x-axis, we need to set up an integral representing the volume.
Consider a small vertical strip in region R of thickness ∆x at a distance x from the y-axis. When this strip is rotated about the x-axis, it sweeps out a small cylindrical disk of thickness ∆x and radius y.
The radius of this disk can be determined by the difference between the functions that bound R: y = 9√(1 - x^2/9) (the top function) and y = 9 - 3x (the bottom function).
So, the radius of the disk is: r = 9√(1 - x^2/9) - (9 - 3x) = 3x - x^2/3.
The volume of this disk is given by: V = π(r^2)∆x.
To find the total volume, we need to integrate the volume of these disks from x = 0 to x = 3, using the formula for the radius we derived:
∫[0 to 3] π(3x - x^2/3)^2 dx.
c. Volume of the Solid Generated when R is Rotated about the y-axis:
Similarly, to find the volume of the solid generated when region R is rotated about the y-axis, we need to set up an integral representing the volume.
Consider a small horizontal strip in region R of thickness ∆y parallel to the x-axis. When this strip is rotated about the y-axis, it sweeps out a small cylindrical disk of thickness ∆y and radius x.
The radius of this disk can be determined by the difference between the functions that bound R: y = 9√(1 - x^2/9) (the top function) and y = 9 - 3x (the bottom function).
So, the radius of the disk is: r = 9 - 3x - 9√(1 - x^2/9).
The volume of this disk is given by: V = π(r^2)∆y.
To find the total volume, we need to integrate the volume of these disks from y = 0 to y = 9, using the formula for the radius we derived:
∫[0 to 9] π(9 - 3x - 9√(1 - x^2/9))^2 dy.
These integrals represent the area and volumes of region R and the solid generated when R is rotated about the x-axis and y-axis.