calculus
posted by Taeyeon on .
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 xaxis. 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 yaxis. Express the integrand as a function of a single variable.

First, see where the curves intersect. Just substitute y=93x 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] y1y2 dx
= ∫(93x)  sqrt(93x^2) dx
(b) volume using discs is
∫pi*(R^2r^2) dx
where R = 93x and r=sqrt(93x^2)
= pi∫(93x)^2  (93x^2) dx
(c) same idea, different axis
pi∫(3  y/3)^2  (3  y^2/3) dy