Calculus Volume 3 · free preview
Double Integrals in Polar Coordinates
Polar double integrals replace the area element with r dr d theta, the Jacobian r accounting for the wedges' growing area, and simplify regions and integrands with circular symmetry.
1. The polar area element
The polar area element
2. When polar wins
When polar wins
3. Setting up polar limits
Setting up polar limits
4. The Jacobian's general theory
The Jacobian's general theory
5. Polar area element
The polar double integral's area element dA equals which expression?
6. When polar wins
A double integral is most profitably converted to polar coordinates when which feature holds?
7. Gaussian plane integral
The integral of e^(-(x^2 + y^2)) over the whole plane equals which value?
8. Disk area by polar integral
The double integral of 1 over the disk of radius a, in polar coordinates, equals which value?
9. Commonest polar error
What is the commonest error in setting up polar double integrals?
10. Jacobian power
The polar Jacobian is r to which power? Answer with a number.
11. Annulus limits inner
The annulus between radii 1 and 2 has r ranging from 1 to 2. As a number, what is the inner limit? Answer with a number.
12. Gaussian radial value
The radial integral of r e^(-r^2) from 0 to infinity equals what number? Answer with a fraction like 1/2.
13. Cartesian versus polar effort
The integral of x^2 + y^2 over the unit disk: the Cartesian setup needs what that polar avoids?
14. Circle r equals two cos theta
The polar curve r = 2 cos theta is a circle of radius 1 centered where? Answer like (1, 0).
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