the integral ∫g(x)d(x) represents the area enclosed by the ellipse (x/7)^2 + (y/2)^2 = 1 what is g(x)?

Question

its a definite integral from 0 to 7

Answer 1

An equation of the ellipse is x² / 49 + y² /4 = 1

The ellipse has center (0,0), x-intercepts (7,0) and (-7, 0) and y-intercepts (0, 2) and (0,-2).

Solving for y, we get y² / 4 = 1 - x² / 49

So, y² = 4 - 4x² / 49

y = ±√(4 - 4x² / 49) = ±2√(1 - x² / 49)

The portion of the curve that lies in the first quadrant has equation y = 2√(1 - x²/49), where 0 < x < 7. The area of the region bounded by this curve, the x-axis and y-axis is 1/4 of the total area of the ellipse.

So, integrating g(x) = 4y = 8√(1 - x² / 49) from x = 0 to x = 7 gives the area of the ellipse.