Chapter 9: Q. 26 (page 775)

The arc length of polar functions: Find the arc lengths of the following polar functions.
$r = \sin θ + \cos θ for θ \in [0, π]$

Short Answer

Expert verified

The arc length is $2 \sqrt{2} π units$

Step by step solution

Given information.

The curve is $r = \sin θ + \cos θ for θ \in [0, π]$

Arc length

The arc length of the curve $r$ is given by,

localid="1663046444567" $L = \int_{α}^{β} \sqrt{r^{2} + {(\frac{d r}{d θ})}^{2}} d θ L = \int_{0}^{2 π} \sqrt{{(\sin θ + \cos θ)}^{2} + {(\cos θ - \sin θ)}^{2}} d θ [∵ \frac{d r}{d θ} = \cos θ - \sin θ] L = \int_{0}^{2 π} \sqrt{(\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ) + (\sin^{2} θ + \cos^{2} θ - 2 \sin θ \cos θ)} d θ L = \int_{0}^{2 π} \sqrt{(1 + 2 \sin θ \cos θ) + (1 - 2 \sin θ \cos θ)} d θ L = \int_{0}^{2 π} \sqrt{2} d θ L = \sqrt{2} \int_{0}^{2 π} d θ L = \sqrt{2} \times 2 π L = 2 \sqrt{2} π units$

Thus, the arc length is $L = 2 \sqrt{2} π units$

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The arc length of polar functions: Find the arc lengths of the following polar functions.
$r = \sin θ + \cos θ for θ \in [0, π]$

Short Answer

Step by step solution

Given information.

Arc length

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The arc length of polar functions: Find the arc lengths of the following polar functions.r=sinθ+cosθforθ∈[0,π]

Short Answer

Step by step solution

Given information.

Arc length

One App. One Place for Learning.

Most popular questions from this chapter

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The arc length of polar functions: Find the arc lengths of the following polar functions.
$r = \sin θ + \cos θ for θ \in [0, π]$