Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Working in polar coordinates, calculate the area within the inner loop of the limaçon r=1/2−cosθ.
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/314−cosθ+1+cos2 θ2 ⅆθ
=12∫−π/3π/334−cosθ+cos2 θ2 ⅆθ
Expression palette: Definite Integral template
Press the Enter key.
Context Panel: Approximate≻10 (digits)
→at 10 digits
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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