Estimate the area of the ellipse given by the equation 4x2+y2=25 as follows:?

Question

The area of the ellipse is 4 times the area of the part of the ellipse in the first quadrant ( x  and y  positive). 

Estimate the area of the ellipse in the first quadrant by solving for y  in terms of x.  Estimate the area under the graph of y  by dividing the interval [0,2.5]  into four equal subintervals and using the left endpoint of each subinterval. Be sure you draw a picture. 

Don't forget to multiply your estimate for the area of the part of the ellipse in the first quadrant by 4 to get the entire area.

kb · Accepted Answer

For the first quadrant area approximation:

Solving for y yields f(x) = y = sqrt(25 - 4x^2).

Δx = (2.5 - 0)/4 = 2.5/4 = 25/40 = 5/8.
So, the interval is subdivided into {0, 5/8, 10/8, 15/8, 20/8 = 2.5}.

Using left endpoints yields
A ≈ Δx [f(0) + f(5/8) + f(10/8) + f(15/8)] 
   = (5/8) [5 + sqrt(375/16) + sqrt(75/4) + sqrt(175/16)] ≈ 10.924.

So, the area of the ellipse is approximately 4(10.924) ≈ 43.696.

(For the record, the exact area is π * (5/2) * 5 ≈ 39.270.
This is reasonable, since left endpoints over-estimate the area in the first quadrant.)

I hope this helps!

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Estimate the area of the ellipse given by the equation 4x2+y2=25 as follows:?

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