- Compute the integral $\ds \int x \sin (x^2)dx$. Pass the chalk (PTC).
- Consider the curve $C$ parametrized by $\vec r(t) = (3-2t^2,4t+5)$ for $-1\leq t\leq 3$.
- Give a vector equation of the tangent line to the curve at $t=2$. PTC
- Recall the arc length formula is $$\int_C ds = \int_a^b\left|\dfrac{d\vec r}{dt}\right|dt=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt.$$ Set up a formula to compute the length of this curve. Just set it up. PTC
- Consider the curve $C$ parametrized by $\vec r(t) = (t^2, t^3)$ for $0\leq t\leq 2$.
- Give a vector equation of the tangent line to the curve at $t=1$. PTC
- Find the length of this curve. Actually compute the integral. PTC
- A wire lies along the curve $C$. The density (mass per length) of the wire at a point $(x,y)$ on the curve is given by $\delta(x,y) = y+2$. Set up an integral formula that gives the total mass of the wire. PTC
- The wire contains charged particles. The charge density (charge per length) at a point $(x,y)$ on the curve is given by the product $q(x,y)=xy$. Set up an integral formula that gives the total charge on the wire.
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