Calculus by OpenStax

By OpenStax

Calculus is designed for the common - or three-semester normal calculus direction, incorporating cutting edge positive aspects to reinforce scholar studying. The booklet courses scholars in the course of the center recommendations of calculus and is helping them know the way these options practice to their lives and the realm round them. as a result complete nature of the fabric, we're delivering the booklet in 3 volumes for flexibility and potency. quantity three covers parametric equations and polar coordinates, vectors, services of a number of variables, a number of integration, and second-order differential equations.

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2 Chapter 1 | Parametric Equations and Polar Coordinates 37 n s = n lim ∑ sk →∞ k=1 ⎛ n 2 2⎞ ^ ⎜ ∑ ⎛⎝x′ ⎛⎝ t k⎞⎠⎞⎠ + ⎛⎝y′ ⎛⎝˜t k⎞⎠⎞⎠ ⎟Δt = n lim →∞ ⎠ ⎝k = 1 =∫ b a (x′ (t)) 2 + ⎛⎝y′ (t)⎞⎠ 2dt. ^ When taking the limit, the values of t k and ˜t k are both contained within the same ever-shrinking interval of width Δt, so they must converge to the same value. We can summarize this method in the following theorem. 3: Arc Length of a Parametric Curve Consider the plane curve defined by the parametric equations x = x(t), y = y(t), t1 ≤ t ≤ t2 and assume that x(t) and y(t) are differentiable functions of t.

A. ⎛ π⎞ ⎝2, 4 ⎠ b. ⎛ 2π ⎞ ⎝−3, 3 ⎠ c. ⎛ 5π ⎞ ⎝4, 4 ⎠ Solution The three points are plotted in the following figure. 29 Three points plotted in the polar coordinate system. 11 ⎛ ⎞ ⎛ ⎞ Plot ⎝4, 5π ⎠ and ⎝−3, − 7π ⎠ on the polar plane. 3 2 Polar Curves Now that we know how to plot points in the polar coordinate system, we can discuss how to plot curves. In the rectangular coordinate system, we can graph a function y = f (x) and create a curve in the Cartesian plane. In a similar fashion, we can graph a curve that is generated by a function r = f (θ).

We can modify the arc length formula slightly. First rewrite the functions x(t) and y(t) using v as an independent variable, so as to eliminate any confusion with the parameter t: x(v) = 140v, y(v) = −16v 2 + 2v. 2 Chapter 1 | Parametric Equations and Polar Coordinates 39 Then we write the arc length formula as follows: s(t) = ∫ =∫ ⎛dy ⎞ ⎛dx ⎞ ⎝ ⎠ + ⎝dv ⎠ dv 0 dv t t 0 2 2 140 2 + (−32v + 2) 2dv. The variable v acts as a dummy variable that disappears after integration, leaving the arc length as a function of time t.

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