Short Answer 
To calculate the area bounded by the ellipse defined by 9x¬≤ + 25y¬≤ = 225 and the x-axis, we first rearranged the equation to x¬≤/25 + y¬≤/9 = 1. We set up the integral to find the area above the x-axis from x = -2 to x = 2, which we solved to find the area is (5 Р‚ao21) square units.
Step 1: Rearranging the Ellipse Equation
To begin calculating the area bounded by the ellipse and the given lines, we first need to rearrange the equation of the ellipse (9x² + 25y² = 225). This can be simplified by dividing every term by 225, resulting in the equation (x²/25 + y²/9 = 1). This representation indicates that we have an ellipse centered at the origin with its major axis along the y-axis, as the larger denominator corresponds to the y² term.
Step 2: Setting Up the Integral
Next, we will set up the integral to calculate the area between the ellipse and the x-axis from (x = -2 to x = 2). The upper half of the ellipse can be represented mathematically as (y = (3/5)‚ao(225 Р9x¬≤)). We will integrate this function to cover the range of x-values, and to find the total area, we will use the integral:
- A = 2 ‚a´(from 0 to 2) (3/5)‚ao(225 Р9x¬≤) dx
Step 3: Solving the Integral
Finally, we will solve the integral using the substitution method. By letting (u = 225 Р9x¬≤), we derive the new limits of integration and change the differential. The integral transforms, allowing us to evaluate it effectively. Ultimately, we find the area bounded by the ellipse and the specified lines to be (5 Р‚ao21) square units, concluding the calculation.