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Home Multiple Integrals Double Integrals in Polar Coordinates Examples Example 1
 
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Example 1

Find the volume over the unit circle x2 + y2 ≤ 1 between the surfaces z = 0 and z = x2.

Step 1

Sketch D and the solid, as in Figure 12.5.7.

12_multiple_integrals-266.gif12_multiple_integrals-267.gif

Figure 12.5.7

Step 2

D is the polar region 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1.

Step 3

12_multiple_integrals-268.gif

For comparison let us also work this problem in rectangular coordinates. We can see that it is easier to use polar coordinates.

D is the region - 1 ≤ x ≤ 1, 12_multiple_integrals-269.gif

12_multiple_integrals-270.gif

We make the trigonometric substitution shown in Figure 12.5.8:

x = sin φ, 12_multiple_integrals-271.gif, dx = cos = φ dφ.

12_multiple_integrals-273.gif

Figure 12.5.8

Then φ = -π/2 at x = -1 and (φ = π/2 at x = 1, so

12_multiple_integrals-272.gif

12_multiple_integrals-274.gif

 

Home Multiple Integrals Double Integrals in Polar Coordinates Examples Example 1

Last Update: 2006-11-15