L 25 Integration in Physics 1 By

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L 25 – Integration in Physics 1 By the end of this lecture you should be able to: • Know the Importance of Integration in Understanding Physics • Find the Area of simple Geometric Shapes using integration • Find the Total Mass and Center of Mass (centroid) of simple Geometric Shapes • Set up Formulas for other physical examples of changing characteristics ( mass density, charge density, kinetic energy, etc…) and • Use these formulas in integrals to solve for the unknowns

Area and Volume Elements of Simple Geometric Objects With Central and Axial Symmetry Area of a Circle R r dr d. A = ______ Total area A?

Finding Volume of a Cylinder with Length L and Radius R R dr L r d. V = ______ Total volume V?

Finding Volume of a Sphere R r Total volume V? d. V = ______ dr

Finding masses Exercise 1: The mass density of an axially symmetric solid cylinder L of height L and radius R changes as follows: Find the total mass of the cylinder. R

Finding masses Exercise 2: The mass density of a centrally symmetrical spherical shell is given by: for Find the total mass of the sphere if the inner and outer radii are R 1 and R 2 respectively.

Finding moments of inertia Moment of inertia of a point mass Axis of rotation m r Moment of inertia of a mass element Axis of rotation dm r

Finding moments of inertia Exercise 3: Find the moments of inertia of a uniform rod of mass M and length L about axis A and B: A B Equal segments = L/6

Exercise 4: The mass density of an axially symmetric solid cylinder of height L and radius R changes as follows: R L The cylinder is rotating with an angular frequency w about its symmetry axis. Find the rotational kinetic energy of the cylinder.

Example 5: The surface charge density of a strip of width W = 0. 010 m and length L = 0. 200 m is increasing linearly from one end to the other as follows: W x L where a = 5. 00 m. C/m 3. What is the total charge of the strip?

Additional Resources • http: //www. intmath. com/applicationsintegration/1 -apps-indefinite-integral. php • http: //hyperphysics. phyastr. gsu. edu/hbase/hframe. html A general site that can help you find specific example from many different areas of physics

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