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Chapter 9: Problem 49

Give the integral formulas for area and arc length in polar coordinates.

Short Answer

Expert verified
The area of a polar curve \( r = f(胃) \) is given by the integral formula \( A = \frac{1}{2}\int_{伪}^{尾} [f(胃)]^2 d胃 \). The arc length of the same polar curve is given by the integral formula \( s = \int_{伪}^{尾} \sqrt{[f(胃)]^2 + [f'(胃)]^2} d胃 \).

Step by step solution

01

State the Area Integral Formula in Polar Coordinates

The formula for the area \( A \) of a polar curve \( r = f(胃) \), for \( 胃 \) between alpha (伪) and beta (尾), is given by: \[ A = \frac{1}{2}\int_ {伪}^{尾} [f(胃)]^2 d胃 \] This formula determines the area bound by the curve and the rays from the origin making angles 伪 and 尾 with the positive x-axis.
02

State the Arc Length Integral Formula in Polar Coordinates

The formula for the arc length \( s \) of a polar curve \( r = f(胃) \), for \( 胃 \) between alpha (伪) and beta (尾), is given by: \[ s = \int_{伪}^{尾} \sqrt{[f(胃)]^2 + [f'(胃)]^2} d胃 \] This formula calculates the length of the curve traced out as \( 胃 \) varies from 伪 to 尾.

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