using geometry, calculate volume of solid under z= square root of 49-x^2-y^2 and over circular disk x^2+y^2 is less than or equal to 49

To calculate the volume of the solid under the surface z = √(49 - x^2 - y^2) and over the circular disk x^2 + y^2 ≤ 49, we can use a triple integral in cylindrical coordinates.

First, we need to describe the limits of integration for each variable.

In cylindrical coordinates, x = r*cos(θ) and y = r*sin(θ). The circular disk x^2 + y^2 ≤ 49 can be expressed as r^2 ≤ 49. Since r represents the distance from the origin to a point in the disk, the limits for r will be from 0 to √(49) = 7.

Next, we need to determine the limits for the angle θ. The circular disk is symmetric around the z-axis, so we can integrate over the entire circle. The limits for θ will be from 0 to 2π.

Finally, to calculate the height or z-coordinate, we use z = √(49 - x^2 - y^2). Substituting x = r*cos(θ) and y = r*sin(θ), we get z = √(49 - r^2*cos^2(θ) - r^2*sin^2(θ)) = √(49 - r^2).

Therefore, the integral to calculate the volume is:

V = ∫∫∫ (49 - r^2) r dr dθ dz,

with the limits of integration:

r: 0 to 7,
θ: 0 to 2π,
z: 0 to √(49 - r^2).

By evaluating this triple integral, you will obtain the volume of the solid under the given surface and over the circular disk.