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Recall that a median of a triangle is a segment connecting a vertex of a triangle to the midpoint of the opposite side. Let T be the triangle with vertices$(0,0),(a,0),and(c,d).$In Exercises 70鈥�72, prove the given statements.

The medians of triangle T are concurrent; that is, all three medians intersect at the same point, P.

Prove Theorem $13.2\left(a\right)$. That is, prove that$\underset{j=1}{\overset{m}{鈭�}}\underset{k=1}{\overset{n}{鈭�}}{a}_{jk}=\underset{k=1}{\overset{n}{鈭�}}\underset{j=1}{\overset{m}{鈭�}}{a}_{jk}$

Let $a$be a constant. Prove that the equation of the plane $x=a$isrole="math" localid="1652390612497" $蚁=acsc蠒sec胃$ in spherical coordinates.

$$\begin{gathered} Let\mathfrak{R}=\left\{\left(x,y,z\right)|a_1猢�x猢�a_2,b_1猢�y猢�b_2,c_1猢�z猢�c_2\right\}.Provethat: \\ {鈭�}_{\mathfrak{R}}dV=\left(a_2-a_1\right)\left(b_2-b_1\right)\left(c_2-c_1\right). \\ Whatistherelationshipbetween\mathfrak{R}andtheproduct\left(a_2-a_1\right)\left(b_2-b_1\right)\left(c_2-c_1\right). \end{gathered}$$

Recall that a median of a triangle is a segment connecting a vertex of a triangle to the midpoint of the opposite side. Let T be the triangle with vertices $(0,0),(a,0),and(c,d).$In Exercises 70鈥�72, prove the given statements.

Use the integral definition for the centroid to show that the centroid of T is point P from Exercise 70.

Let $b$be a constant. Prove that the equation of the plane $y=b$is$r=bcsc蠁csc胃$in spherical coordinates.

Let $f\left(x\right)$be an integrable function on the rectangular solid $\mathfrak{R}=\left\{\left(x,y,z\right)|a_1猢�x猢�a_2,b_1猢�y猢�b_2,c_1猢�z猢�c_2\right\}$, and let $魏鈭�\mathrm{鈩�}.$Use the definition of the triple integral to prove that:

${鈭�}_{\mathfrak{R}}魏f(x,y,z)dV=魏{鈭�}_{\mathfrak{R}}f(x,y,z)dV.$

Recall that a median of a triangle is a segment connecting a vertex of a triangle to the midpoint of the opposite side. Let T be the triangle with vertices $(0,0),(a,0),and(c,d).$In Exercises 70鈥�72, prove the given statements.

Prove that the centroid of triangle T is two-thirds of the way from each vertex to the opposite side.

Let $R$be the radius of the base of a cone and $h$be the height of the cone. Use cylindrical coordinates to set up and evaluate a triple integral proving that the volume of the cone is$\frac{1}{3}蟺R^2$.

Let $a<b\text{and}c<d$be real numbers and $R$be rectangle defined by

$R=\left\{\right(x,y)鈭�a鈮�x鈮�b\text{and}c鈮�y鈮�d\}$in $xy$plane. If $g\left(x\right)$is continuous in interval $[a,b]$and $h\left(y\right)$is continuous on $[c,d]$

Use Fubini's theorem to prove that${鈭�}_{R}g\left(x\right)h\left(y\right)dA=\left({鈭�}_a^{b}g\left(x\right)dx\right)\left({鈭�}_c^{d}h\left(y\right)dy\right).$