91Ó°ÊÓ

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=\sqrt{t}, y=t, z=t ; 1 \leq t \leq 6\)

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ x=\ln t, y=3 t, z=t^{2} ; t_{1}=2 $$

Show that the volume of the solid bounded by the elliptic paraboloid \(x^{2} / a^{2}+y^{2} / b^{2}=h-z, h>0\), and the \(x y\) plane is \(\pi a b h^{2} / 2\), that is, the volume is one-half the area of the base times the height. Hint: Use the method of slabs of Section \(6.2\).

Find the length of the curve with the given vector equation. $$ \mathbf{r}(t)=t \cos t \mathbf{i}+t \sin t \mathbf{j}+\sqrt{2 t} \mathbf{k} ; 0 \leq t \leq 2 $$

Find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{i}} \mathbf{u}\)

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=t, y=t^{2}, z=t^{3} ; 1 \leq t \leq 2\)

If both \(\mathbf{u} \times \mathbf{v}=\mathbf{0}\) and \(\mathbf{u} \cdot \mathbf{v}=0\), what can you conclude about u or \(\mathbf{v}\) ?

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=2 \cos t, y=\sin t, z=t ; 0 \leq t \leq 6 \pi\)

Find the length of the curve with the given vector equation. $$ \mathbf{r}(t)=\sqrt{6} t^{2} \mathbf{i}+\frac{2}{3} t^{3} \mathbf{j}+6 t \mathbf{k} ; 3 \leq t \leq 6 $$

A 100 -pound chandelier is held in place by four wires attached to the ceiling at the four corners of a square. Each wire makes an angle of \(45^{\circ}\) with the horizontal. Find the magnitude of the tension in each wire.