Geometric application: arc length Problem: Find the length s of the arc defined by the curve y=f(x) from a to b. Solution: Use differential element method, take any element [x,x+dx], the length of the sub-arc corresponding to [x,x+dx] is: Therefore the total arc length is

Arc length formula If a curve has the equation x=g(y), then the length of the arc corresponding is Ex. Find the length of the arc of the parabola from (0,0) to (1,1). Sol.

Differential of arc length function A smooth curve is defined by y=f(x) and let s(x) be the length of the arc corresponding to the part from a to x. Then s(x) is called the arc length function, and we have formula The differential of arc length is

Geometric application: surface area A surface of revolution is generated by rotating a planar curve about a line. Problem: find the area of the surface obtained by rotating the curve about the x-axis.

Surface area formula Use differential element method, take an element [x,x+dx], the small surface is approximately a portion of a circular cone, thus its surface area is and therefore the total surface area is

Surface area formula If the curve is given by and rotate the curve about y-axis, then the surface area formula is Ex. The curve about x-axis, find the surface area. Sol.

Surface area formula If the curve is given by and rotate it about x-axis, the surface area formula is If the curve is and rotate about y-axis, the formula is

Example Ex. Find the area of the surface generated by rotating the curve segment from (1,1) to (2,4), about the y-axis. Sol I. Sol II.

Physical application: hydrostatic pressure Background: when an object is submerged in a fluid, the fluid will exert a force upon the object. The hydrostatic pressure is defined to be the force per unit area. Suppose the object is a thin horizontal plate. Its area is A and is submerged into depth d of a fluid with density Then we have By the principle of fluid pressure, at any point in a liquid the pressure is the same in all directions, we have

Hydrostatic force Ex. A cylindrical drum with radius 3m is submerged in water 10m deep. Find the hydrostatic force on one end of it. Sol. Use differential element method. First build up the coordinate frame with origin placed at the center of the drum. Take any infinitesimal element [x,x+dx], the pressure on this part is and the force is Therefore the total force is

Moments and centers of mass Center of mass: a point on which a thin plate balances horizontally A system of two masses and which lie at and respectively, the center of mass of the system is and are called the moments of the masses and (with respect to the origin) respectively.

Moment and center of mass A system of n particles located at the points on the x-axis, then is the center of mass of the system, is the total mass of the system, and the sum of individual moments is called the moments of the system about the origin.

Moment and center of mass A system of n particles located at the points in the xy-plane, then we define the moment of the system about the y-axis to be and the moment of the system about the x-axis as the coordinates of the center of mass of the system where is the total mass.

Moment and center of mass Consider a flat plate (called a lamina) with uniform density that occupies a region R of the plane. Center of mass is called the centroid The symmetry principle says that if R is symmetric about a line l, then the centroid of R lies on l. The moments are defined so that if the entire mass of a region is concentrated at the center of mass, then its moments remain unchanged. The moment of the union of two nonoverlapping regions is the sum of the moments of the individual regions.

Moment of a planar region Suppose the region R lies between the lines x=a and x=b, above the x-axis and below the curve y=f (x). Use differential element method: the moment of the subregion corresponding to [x,x+dx], about y-axis, is So the moment of R about y-axis is similarly, the moment of R about x-axis is

Center of mass of a planar region The center of mass of R is defined by the total mass is so the coordinates of the centroid are

Example Ex. Find the centroid of the region bounded by the curves y=cosx, y=0, x=0, and Sol.

Center of mass of a planar region Suppose now the region R lies between the lines x=a and x=b, above the curve y=g(x) and below the curve y=f (x) where The coordinates of the centroid are

Example Ex. Find the centroid of the region bounded by the line y=x and Sol.

Homework 20 Section 7.8: 25, 37, 40, 54, 55, 56, 58, 77 Section 8.1: 20, 37