The formula for the surface area of the figure formed by revolving the positive function y = f(x) from x = a to x = b around the x-axis is given by the formula SA = 2n f(x)/1+ f'(x)² dr. The figure formed 1 by revolving y = – for x >1 around the x-axis is called Gabriel's horn or Torricelli's trumpet. Find the volume and surface area of this figure.

At the bottom of a swimming pool, the pressure exerted at a given point is proportional to the depth of the water above it. In particular, the pressure at a given point of depth d is equal to 10, 000d Pascals. We will investigate the total pressure at the bottom of a swimming pool depending on its shape. You may view 3D models of the swimming pools for reference here (note that this is one model plotting the depth as a function of x and y, so represents essentially the pool flipped upside-down): https://www.geogebra.org/m/nnf4yhax (a) Suppose we have a rectangular pool that is 50 meters long and 25 meters wide. The pool has a shallow end that is 1 meter deep at the edge, a deep end 50 meters away that is 3 meters deep at the edge, and is such that the bottom of the pool forms a linear slant in between (so along a line segment parallel to the short side, the pool has constant depth, and along a line segment parallel to the long side, the pool's depth varies linearly). Set up and…

Use linearization to estimate the change in volume of a sphere when its radius changes from r to r+ Δr. Then calculates the exact value of the volume change and calculates the difference of both results