Many times the relationship involving the rate of change of two variables is known, but we may need to find the direct relationship between the two variables. For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time. To find this direct relationship, we need to use integration or anti-differentiation which is opposite to differentiation.

Over 2000 years ago, Archimedes found formulas for the surface areas and volumes of solids such as the sphere, and the cone. Later Leibniz and Newton discovered calculus with the idea that differentiation and integration can undo each other. Then Gauss created the first table of integrals, and many continued to apply integrals mathematically. Riemann and Lebesgue created a logical explanation for integration. Then in 1969 Risch created a general theory and practice of integrating functions.

Integration is a way to determine the anti derivative or a way to undo the derivative process.

The Sydney Opera House has a design based on slices out of a ball. Many differential equations (one type of integration) were solved to create the design of this building.

Historically, one of the first applications of integration was in finding the volumes of wine- casks (which have curved surfaces). The Petronas Towers in Kuala Lumpur experience high forces due to the wind.. Integration was used to create this design for strength.

Outer RingInner Ring VOLUME = Note: Use this formula to find the volume of a solid when two equations are given

Rose is celebrating a friend’s birthday on Tuesday and decides to make an angel cake of three layers for the event. If Region A which is bounded by y= 1, x = 0, and y = x 2 and rotated about the x-axis forms a layer of cake, what is the volume of the cake in cubic inches?

Volume= Given: y=1, x=0 y=x 2 1. Find the limits of integration or the x-values for which the 2 equations intersect 1=x 2  x=1 2. Set up the equation 3. Integrate Volume= = (1 - 1/5 ) =2.513 in 3 Since there are 3 layers… x3 = in 3

Note: Use this formula to find the volume of a solid when one equation is given

Peter has to make a metal cone for his engineering class. If Region R in the 1 st quadrant is bounded by y= -x+6, what is the volume of the cone in cubic centimeters?

1.Set up equation Volume= 2. Integrate Volume= 3. Solve Volume= (216/3 – ) = cm 3

When is integration used? Find area under a curve: Basic Formula:

Find area between two curves: Basic Formula: Note: To find the area between two curves take the top function subtract the bottom function.

Determine the original function when the initial condition is given: When given the velocity function the position (original) function could be determined. Example: Velocity function = 3x 2 + 6x – 2, x(0) = 7. Find the position function.

Real Life Example Bob drove around Long Island at 10 a.m. and returned home at 8 p.m. at night. He was driving at a rate modeled by the function Find how many miles Bob drive from 10 a.m. to 8 p.m.?

Pictures.

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