Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: Figure 7.3 Work done by a constant force. If you are redistributing all or part of this book in a digital format, d F d cos, which the figure also illustrates as the horizontal component of the force times the magnitude of the displacement.Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License The width of each subinterval is given by the formula Δ θ = ( β − α ) / n, Δ θ = ( β − α ) / n, and the ith partition point θ i θ i is given by the formula θ i = α + i Δ θ. Our first step is to partition the interval into n equal-width subintervals. For polar curves we use the Riemann sum again, but the rectangles are replaced by sectors of a circle.Ĭonsider a curve defined by the function r = f ( θ ), r = f ( θ ), where α ≤ θ ≤ β. Recall that the proof of the Fundamental Theorem of Calculus used the concept of a Riemann sum to approximate the area under a curve by using rectangles. Now we turn our attention to deriving a formula for the area of a region bounded by a polar curve. We have studied the formulas for area under a curve defined in rectangular coordinates and parametrically defined curves. In this section, we study analogous formulas for area and arc length in the polar coordinate system. Similarly, the arc length of this curve is given by L = ∫ a b 1 + ( f ′ ( x ) ) 2 d x. This fact, along with the formula for evaluating this integral, is summarized in the Fundamental Theorem of Calculus. In particular, if we have a function y = f ( x ) y = f ( x ) defined from x = a x = a to x = b x = b where f ( x ) > 0 f ( x ) > 0 on this interval, the area between the curve and the x-axis is given by A = ∫ a b f ( x ) d x. In the rectangular coordinate system, the definite integral provides a way to calculate the area under a curve. 7.4.2 Determine the arc length of a polar curve.7.4.1 Apply the formula for area of a region in polar coordinates.
0 Comments
Leave a Reply. |