Question

The equation x2 16 + y2 9 = 1 defines an ellipse, which is graphed above. in this excercise we will approximate the area of this ellipse. (a) to get the total area of the ellipse, we could first find the area of the part of the ellipse lying in the first quadrant, and then multiply by what factor? 4 correct: your answer is correct. (b) find the function y=f(x) that gives the curve bounding the top of the ellipse.

Answer 1

(a) 4 
(b) y = sqrt(9 - (9/16)x^2)  
The best guess to the formula using knowledge of the general formula for an ellipse is: 
x^2/16 + y^2/9 = 1  
(a). An ellipse is reflectively symmetrical across both the major and minor axis. So if you can get the area of the ellipse in a quadrant, then multiplying that area by 4 would give the total area of the ellipse. So the factor of 4 is correct. 
 (b). The general equation for an ellipse is not suitable for a general function since it returns 2 y values for every x value. But if we restrict ourselves to just the positive value of a square root, that problem is easy to solve. So let's do so:
 x^2/16 + y^2/9 = 1
 x^2/16 + y^2/9 - 1 = 0
 x^2/16 - 1 = - y^2/9
 -(9/16)x^2 + 9 = y^2
 9 - (9/16)x^2 = y^2
 sqrt(9 - (9/16)x^2) = y
 y = sqrt(9 - (9/16)x^2)