Find the area in the first quadrant bounded by the curve y=(9-x)^1/2 and the x- and y-axis.

To find the area in the first quadrant bounded by the given curve, we can use integration techniques. Let's break down the steps to calculate the area:

Step 1: Identify the limits of integration.
Since we are working in the first quadrant, the x-values range from 0 to 9. Therefore, the limits of integration for x would be 0 and 9.

Step 2: Set up the integral.
We can calculate the area by integrating the equation of the curve with respect to x. In this case, the equation of the curve is y = (9 - x)^(1/2). So, we need to integrate this equation from x = 0 to x = 9.

Step 3: Integrate the equation.
To integrate the equation y = (9 - x)^(1/2) with respect to x, we can rewrite it as y^2 = 9 - x and solve for x: x = 9 - y^2.
Thus, the integral becomes ∫(9 - y^2) dx.

Step 4: Evaluate the integral.
The integral of (9 - y^2) with respect to x is x(9 - y^2) + C, where C is the constant of integration. We can evaluate the integral over the limits of integration, which are from 0 to 9:
Area = ∫(0 to 9) (9 - y^2) dx = [x(9 - y^2)](0 to 9)

Substituting the limits of integration:
Area = [9(9 - y^2)] - [0(9 - y^2)]
Area = 81 - 9y^2

Step 5: Simplify the expression.
The expression 81 - 9y^2 represents the area bounded by the curve y = (9 - x)^(1/2), the x-axis, and the y-axis in the first quadrant.

Therefore, the area in the first quadrant bounded by the curve y = (9 - x)^(1/2) and the x- and y-axis is 81 - 9y^2.