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angle antiderivative apply approximation axis calculator centroid chain rule circle compute constant continuous function converges critical points curve cylinder decimal places deﬁned deﬁnition denote distance diverges double integral ellipse endpoints evaluate Example ﬁgure ﬁnd Find the volume ﬁrst ﬁxed ﬂow formula function f given gives graph height Hence horizontal hyperbola improper integral inﬁnite series intersection length limit line integral logarithm maximum mean value theorem method minimum value normal vector obtain open interval parabola paraboloid parametric parametric curve partial derivatives perpendicular plane polynomial positive Problem proof radius real number rectangle rectangular result revolving Riemann sum satisﬁes sequence shown in Fig slope solid Solution sphere spherical square subintervals substitution sufﬁciently Suppose surface area tangent line tank Theorem tion triangle trigonometric trigonometric functions variable vector ﬁeld velocity vertical x-axis xy-plane y-axis yields zero