Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Example 7.2.6
Working in polar coordinates, calculate the area within the inner loop of the limaçon r=1/2−cosθ.
Solution
Mathematical Solution
According to Figure 7.1.6(a), the top half of the inner loop of this limaçon is traced for θ∈5 π/3,2 π. The full inner loop is traced for θ∈5 π/3,7 π/3, or equivalently, θ∈−π/3,π/3. (Indeed, this is the hardest part of the calculation, determining the range of θ for which the inner loop is drawn.) Of course, the area of the top half of the inner loop could be computed and doubled to get the area of the full inner loop.
Applying the appropriate formula from Table 7.2.1, the area is computed as follows.
12∫−π/3π/31/2−cosθ2 ⅆθ
=12∫−π/3π/314−cosθ+cos2θ ⅆθ
=12∫−π/3π/314−cosθ+1+cos2 θ2 ⅆθ
=12∫−π/3π/334−cosθ+cos2 θ2 ⅆθ
=1234−sinθ+sin2 θ4−π/3π/3
=1234⋅2 π3−32−−32+1432−−32
=π4−338
≐0.13586
Maple Solution
Interactive solution
Expression palette: Definite Integral template Press the Enter key.
Context Panel: Approximate≻10 (digits)
12∫−π/3π/31/2−cosθ2 ⅆθ
14π−383
→at 10 digits
0.1358791105
A stepwise solution can be implemented with the tutor, wherein declaring the Sum and Constant Multiple rules as Understood Rules proves useful.
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