Math Department Series &vector analysis Math205 Science collage
King Abdul Aziz University Dr. Najwa Joharji
Objectives of Chapter 15 Multiple Integrals 15.1 Double Integral:
1. Define the volume using double integral.
2. Define the volume using double integral.
3. Define Fubini's Theorem for double integral over a rectangle
4. Use Fubini's Theorem to solve the double integral over a rectangle easier.
5. Find the volume of the solid under surface z = f (x,y) using Double integral 6. Evaluate a double integral over a rectangular region.
7. Define the double integral over a general region with its two types.
8. Find limits of integration.
9. Evaluate the double integral over a general region.
10. Reverse the order of integration.
11. Verify the properties of double integral.
15.2 Area,Moments,and Centers of Mass:
1. Find the area using double integral.
2. Find the average value using the double integral.
3. Find the mass for a thin plate covering a region R.
4. Find the first moment formula for a thin plate covering a region R.
5. Find the center of mass for a thin plate covering a region R.
6. Find the moments of inertia for a thin plate in the xy-plane.
7. Find Radii of Gyration.
8. Find the center of mass when the density is constant (The centroid).
15.3 Double Integral in Polar Form:
1. Find the area using double integral in polar coordinates.
2. Change Cartesian integrals into polar integrals.
3. Find limit of integration in polar form.
4. Evaluate integral using Polar Coordinates.
5. Use the double integral in polar coordinates to find the volume under z f(r,). 6. Use the double integral in polar coordinates to find the center of mass and
moments.
15.4 Triple Integral in Rectangular coordinates:
1. Define the volume using triple integral.
2. Calculate the triple integral over a rectangular box.
3. Define Fubini's Theorem for triple integral over a rectangular box.
Math Department Series &vector analysis Math205 Science collage
King Abdul Aziz University Dr. Najwa Joharji
4. Find the limits of integration in the 3-dimensional space.
5. Use the triple integral to find the volume under a surface z= f(x,y).
6. Calculate the triple integral over a general region.
7. Find the average value of a function space.
8. Verify properties of Triple integral.
15.5 Masses and Moments in Three Dimensions
1. Find the mass for solid objects in space.
2. Find the first moment formula for solid objects in space.
3. Find the moments of Inertia for solid objects in space about the coordinate axes.
4. Find the moments of Inertia for solid objects in space about a line L.
5. Find Radii of Gyration about a line L.
6. Find the center of mass for solid objects in space.
15.6 Triple Integral in Cylindrical and Spherical Coordinates:
1. Define the Cylindrical Coordinates.
2. Transverse from Cylindrical Coordinates to Rectangular Coordinates and vise versa.
3. Determine the cases that we need to solve the triple integral in cylindrical coordinates.
4. Find limits of integrations in Cylindrical Coordinates.
5. Convert the triple integral from rectangular coordinates to cylindrical coordinates.
6. Integrate in Cylindrical Coordinates.
7. Find the volume of a solid in Cylindrical Coordinates.
8. Find the center of mass, moments in Cylindrical Coordinates.
9. Define Spherical Coordinates.
10. Transverse from Spherical Coordinates to Rectangular Coordinates and vise versa.
11. Find a Spherical Coordinates equation for a solid in space given by Rectangular Coordinates.
12. Determine the cases that we need to solve the triple integral in Spherical Coordinates.
13. Convert the triple integral from rectangular coordinates to Spherical Coordinates.
14. Integrate in Spherical Coordinates.
15. Find the volume of a solid in Spherical Coordinates.
16. Find the center of mass, moments in Spherical Coordinates.
