To find the total area between the curve y=25-x^2 and the x-axis from 0 to 6, we need to find the definite integral of the function from 0 to 6. This process is known as integration.
Step 1: Identify the function
The function given is y = 25 - x^2. In order to find the area between this equation and the x-axis, we will need to find the integral of this function.
Step 2: Determine the limits of integration
The area we are interested in lies between x=0 and x=6, so these are our limits of integration.
Step 3: Integrate
We will integrate the function 25 - x^2 from 0 to 6. The antiderivative of 25 - x^2 is 25x - (x^3)/3.
Step 4: Evaluate the antiderivative at the limits of integration
This gives us F(6) - F(0).
F(6) = 25*6 - (6^3)/3 = 150 - 72 = 78
F(0) = 25*0 - (0^3)/3 = 0
So the integral from 0 to 6 of the function 25 - x^2 is F(6) - F(0) = 78 - 0 = 78.
Therefore, the total area between the curve y=25-x^2 and the x-axis from 0 to 6 is 78 square units.