# calculus

posted by .

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.

• calculus -

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

## Similar Questions

1. ### Calculus

Use the shell method to set up, but do not evaluate, an integral representing the volume of the solid generated by revolving the region bounded by the graphs of y=x^2 and y=4x-x^2 about the line x=6. I had the shell radius as (6-x) …
2. ### Calculus AB

Let R be the region bounded by the graphs of y=sin(pi x) and y=(x^3)-4x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line y=-2, x=2, x=-1. Write, but do not evaluate, an integral expression for the area.
3. ### calculus

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. …
4. ### calculus

2. Sketch the region in the first quadrant that is bounded by the graphs of y=x^3, y=4X , and 2x+y-3=0 that lies below both straight lines. Find the area of this region. I can see the graph in front of me. which function do I subtract …
5. ### calculus

1. Find the area of the region bounded by f(x)=x^2 +6x+9 and g(x)=5(x+3). Show the integral used, the limits of integration and how to evaluate the integral. 2. Find the area of the region bounded by x=y^2+6, x=0 , y=-6, and y=7. Show …
6. ### Calculus

Let f be the function given by f(x)=(x^3)/4 - (x^2)/3 - x/2 + 3cosx. Let R be the shaded region in the second quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line l, the line tangent …
7. ### calc 3

1. Evaluate the given integral by making an appropriate change of variables, where R is the region in the first quadrant bounded by the ellipse 36x^2+25y^2=1. L= double integral R (4sin(144x^2+100y^2) dA. 2. Use the given transformation …
8. ### Calculus check

The functions f and g are given by f(x)=sqrt(x^3) and g(x)=16-2x. Let R be the region bounded by the x-axis and the graphs of f and g. A. Find the area of R. B. The region R from x=0 to x=4 is rotated about the line x=4. Write, but …
9. ### AP Calculus AB

Which integral gives the area of the region in the first quadrant bounded by the axes, y = e^x, x = e^y, and the line x = 4?