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Test 2 - Review
Make sure you memorize all conversions between coordinate systems and also that you have typical derivatives/antiderivatives memorized. Also, remember what dA and dV are in spherical and cylindrical coordinates... they aren’t just differentials.
How to set up a double integral given a region in the xy-plane.
- Sketch the region. I identify largest and smallest x and y values in the region.
- Set up your bounds.
dydx If setting up dydx integration, you need to figure out which func- tion is on bottom and which is on top. Make sure that the same functions are always on top and bottom. If there are multiple functions bounding then you will need to set up multiple inte- grals, or switch to dxdy in hopes that that setup requires only 1 set of integrals.
- Setting up the inner integral: lower bound = bottom function upper bound = upper function
- Setting up the outer integral: lower bound = smallest x-value in region upper bound = largest x-value in region
dxdy If setting up dxdy integration, you need to figure out which func- tion is on the left and which is on the right. Make sure that the same functions are always on the left and the right. If there are multiple functions bounding then you will need to set up multiple integrals, or switch to dydx in hopes that that setup requires only 1 set of integrals.
- Setting up the inner integral: lower bound = left function
upper bound = right function
- Setting up the outer integral: lower bound = smallest y-value in region upper bound = largest y-value in region
How to reverse the order of integration
- Sketch the region of integration.
To do so, you need to match up the bounds on the inner (respectively outer) integral with the inner (respectively outer) variable of integra- tion.
Remember that the inner integral is a function to function calculation and the outer integral is a number to number calculation.
Draw in the bounding functions from the inner integral. Chop them off according to the outer integral bounds. Remember to make sure you have the correct region by checking that your bottom bound is the bottom or left function and the top bound is the top or right function.
- Set up the double integral with the reversed order of integration follow- ing the same steps in the above explanation of how to set up a double integral given a region in the xy-plane ...the region you’re given is the one you drew yourself.
How to set up a polar double integral over a given region in the xy-plane
- Sketch the region just as you would if you were working in rectangular coordinates.
- Identify the functions that bound r. Remember r is a ray that stretches out from the origin (so it can never be negative) through your region.
- Convert bounding functions on r to polar coordinates using the appro- priate conversions.
- Determine the surface to surface bounds for the first variable of inte- gration that you chose.
- Determine the shadow in the plane of the remaining two variables. Remember, the shadow will come from the fattest part of the shape as you shine your flashlight toward the object along the axis of the variable that you did the surface to surface calculation for. So if you choose to integrate with respect to z first, then your flashlight will be held on the z-axis pointing down onto the object. What shadow is cast?
Think about how all of your bounding functions are relating to each other at the fattest part of your shape. Is the fattest part at the intersection of two surfaces? Set them equal to each other and solve. This will help you figure out what to graph in the remaining plane.
- Set up the outer double integral as you would to find the area of the shadow.
How to find mass, moments, and center of mass
For a thin plate:
- To find the center of mass, you’ll need to find M , Mx, and My.
- Set up the double integral that would calculate the area of the region over which you have the thin plate.
- For the mass, make the function inside your double integrals the density function you are given. If the density is constant, make sure you write somewhere “δ = 1”.
- For the moments about the x- and y-axis use the formulas given in your 13.6 notes. The double integral will be the same as that of the mass (i.e. the area double integral necessary for your region).
For a solid:
- To find the center of mass, you’ll need to find M , Mxy, Mxz , and Myz.
- Set up the triple integral that would calculate the volume of the solid.
- For the mass, make the function inside your triple integral the density function you are given. Again, if the density is constant, make sure you write somewhere “δ = 1”.
- For the moments about the 3 different coordinate planes, you will need to use the formulas given in your 13.6 notes. The triple integral will be the same as that of the mass (i.e. the volume triple integral necessary for your solid).
How to find the volume under a function z = f (x, y) over a region in the xy-plane
Set up the double integral that you would to find the area of the region in the xy-plane. Make the function you’re integrating the function f (x, y).
How to set up a triple integral in cylindrical coordinates to find the volume of a solid D
- Draw the region just as you would if you were setting up a triple integral in rectangular coordinates.
- Figure out the surface to surface calculation for z, i.e. what surface is on the bottom and what surface is on the top. These are your upper and lower bounds of the inner most integral. However, you have to convert them to polar coordinates first. Then put them on the integral.
- Determine the shadow in the xy-plane just as you normally would in rectangular coordinates if you were doing z first.
- The outer double integral is the double integral necessary for finding the area of the shadow in POLAR coordinates.
- Don’t forget that dV = r dzdrdθ.
How to set up a triple integral in spherical coordinates to find the volume of a solid D
- Draw the solid.
- Draw the vertical cross-section slice with your shape on it (it should be a 2D picture).
- If you are asked to change the order of integration or change the co- ordinate system, now you can set up the new triple integrals just as you would as described in the previous explanations above. This is because you now simply have your solid and it is as if you’re starting from scratch on new problem.
