8. This is just the rate of change of the radius with angle. For instance, if you describe a circle, you would expect this derivative to be zero, since the radius does not change with angle, and this is seen by f(θ) = R f′(θ) = 0 f ( θ) = R f ′ ( θ) = 0. The tangent to the curve can be calculated as follows.. Calculate. −x 2y dx + 2 xy dy, where C is the circle of radius 2 centered on the origin. C. Answer: Green's theorem tells us that if F = (M, N) and C is a positively oriented simple closed curve, then.

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Practice 1: When r is given by a formula we can calculate dy/dx , the slope of the tangent line, by using the polar-rectangular conversion formulas dy dy dx and the Chain Rule. By the Chain Rule dθ = dx dθ , so we can dy dx solve for dx by dividing each side of the equation by dθ . dθ dx. dθ.. Example 15.7.3: Setting up a Triple Integral in Two Ways. Let E be the region bounded below by the cone z = √x2 + y2 and above by the paraboloid z = 2 − x2 − y2. (Figure 15.5.4). Set up a triple integral in cylindrical coordinates to find the volume of the region, using the following orders of integration: a. dzdrdθ.