Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Working in polar coordinates, calculate the circumference of the circle r=2 a cosθ.
According to Table 7.2.1, the arc length is obtained by evaluating the integral ∫θ1θ2r2+r′2ⅆθ for r=2 a cosθ. From Figure 7.2.1(a), the diameter of the circle is d=2 a, so the expected value for the arc length is the circumference of the circle, namely, π d=2 a π. This is indeed the case, as the following integration shows.
∫0π2 a cosθ2+ddθ2 a cosθ2 ⅆθ
=∫0π2 a ⅆθ
=2 a π
Expression palette: Definite Integral template and ordinary derivative template
Fill in fields appropriately.
Context Panel: Simplify≻Assuming Positive
∫0π2 a cosθ2+ⅆⅆ θ 2 a cosθ2 ⅆθ→assuming positive2⁢π⁢a
In some instances, Maple considers that a definite integral is made "simpler" by evaluation. Here, the evaluation could have been done with the value command under the aegis of the assumption that a is positive. Without this assumption, Maple will return csgna, that is, the "complex sign" of a.
Applying the Simplify option in the Context Panel avoids the need for the commands illustrated below.
Give the arc length integral the name L.
Append the colon and press the Enter key.
L≔∫0π2 a cosθ2+ⅆⅆ θ 2 a cosθ2 ⅆθ:
Apply the value command with the assuming option.
valueL assuming a>0
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