Integrals with parametric

Unit 8.1 of Math 152

example

• EX1 Find the slope of the tangent to the astroid $x=a{\cos }^3\theta$, $y=a{\sin }^3\theta$ as a function of the parameter $\theta$

  • $\frac{dy}{dx}=\frac{y^{\prime }(\theta )}{x^{\prime }(\theta )}=\frac{3a{\sin }^3\theta \cos \theta }{-3a{\cos }^3\theta \sin \theta }$
    • => $-\tan \theta$

      • At what point is the tangent horizontal

        • $-\tan \theta =0⟹\sin \theta =0$
          • => $\theta =0,\pi$

      • At what point is the tangent vertical

        • $\cos \theta =0⟹\theta =\frac{\pi }{2},\frac{3\pi }{2}$

      • at what point is slope = 1

        • $-\tan \theta =1⟹\theta =\frac{3\pi }{4},\frac{7\pi }{4}$

      • at what point is slope equal to -1

        • $-\tan \theta =-1⟹\theta =\frac{\pi }{4}$

• Ex 2, find the area under one arch of the cycloid

  • Area

    • $x=r(t-\sin t),y=r(1-\cos t),0\le t\le 2\pi$
      • $A={\int }_0^{2\pi }y(x)dx={\int }_0^{2\pi }y(t)x^{\prime }(t)dx$
        • $/{\int }_0^{2\pi }r(1-\cos t)r(1-\cot )dt$
          • => $r^2{\int }_0^{2\pi }1-2\cos t+{\cos }^{2}tdt$
            • => $r^2\left(t-2\sin t+\frac{1}{2}t+\frac{1}{4}\sin 2t\right){|}_0^{2\pi }$
              • => $r^{2}3\pi$

  • Arc length

    • L=${\int }_0^{2\pi }\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}dt$ => ${\int }_0^{2\pi }\sqrt{r^2(1-\cos t{)}^2+r^2{\sin }^{2}t}dt$
      • => $r{\int }_0^{2\pi }\sqrt{1-2\cos t+{\cos }^{2}t+{\sin }^{2}t}dt$ => $r{\int }_0^{2\pi }\sqrt{2-2\cos t}dt$
        • => $r{\int }_0^{2\pi }\frac{2\sin t}{2},dt$
          • = 8t

• Ex 3 Find the surface area

  • $x=3t-t^3$, $y=3t^2$,$0\le t\le 1$
    • $A={\int }_0^{1}2\pi 3t^2\sqrt{(3-3t^2{)}^2+(6t{)}^2}dt$
      • $6\pi {\int }_0^1{t}^2\sqrt{9+9t^4-18t^2+36t^2}dt$
        • => $6\pi {\int }_0^1{t}^2\sqrt{(3+3t^2{)}^2}dt$
          • => $18\pi {\int }_0^1{t}^2+t^{4}dt$
            • =$\frac{48}{5}\pi$

• Ex 4 Polar coordinate are parametric equations

  • $r=1+\cos \theta$
    • $x^{\prime }=r^{\prime }(\theta )\cos \theta -r(\theta )\sin \theta$
    • $y^{\prime }=r^{\prime }(\theta )\sin \theta +r(\theta )\cos \theta$
      • $\sqrt{(r^{\prime }\cos \theta -r\sin \theta {)}^2+(r^{\prime }\sin \theta +r\cos \theta {)}^2}d\theta$
        • => $\sqrt{r^{\prime }{\cos }^2\theta +r^2{\sin }^2\theta -2r{r}^{\prime }\cos \theta \sin \theta +r^{\prime 2}{\sin }^2\theta +r^2{\cos }^2\theta +}$

          • finding the area using the equation

            • $L={\int }_0^{2\pi }(\sqrt{{\sin }^2\theta +(1+\cos \theta {)}^2})d\theta$
              • = 8

          • finding the volume of revolution

            • A=${\int }_0^{\pi }2\pi r(\theta )\sin \theta \sqrt{2+2\cos \theta }d\theta$
              • => $2\pi {\int }_0^{\pi }(1+\cos \theta )\sin \theta \sqrt{2+2\cos \theta }d\theta$
                • => $-2\pi {\int }_2^{0}u\sqrt{2u}du$
                  • =>$\frac{8\pi }{5}$