05 Multiple Integrals Free Online FlashCards

What does a double integral accumulate?

A double integral accumulates a quantity over a planar region, while a triple integral accumulates a quantity throughout a solid region.

What does Fubini’s theorem permit for rectangular regions?

For a continuous function on a rectangle, Fubini’s theorem allows the order of integration to be reversed without changing the integral’s value.

How is a Type II planar region described?

A Type II region uses horizontal slices: $c\le y\le d$ and $h_1(y)\le x\le h_2(y)$.

Evaluate $\int_0^2\int_1^3(x+2y)\,dy\,dx$.

The value is $20$, because $\int_0^2\int_1^3(x+2y)\,dy\,dx=20$.

How is the area of a planar region $D$ expressed?

The area is $\iint_D1\,dA$. Integrating the constant function $1$ counts area throughout the planar region.

How is volume between surfaces computed?

For $f\ge0$, the volume is $\iint_D f(x,y)\,dA$. If two surfaces bound the solid, integrate top minus bottom.

How is the mass of a lamina computed?

The mass is $m=\iint_D\rho(x,y)\,dA$. For constant density $\rho_0$, this becomes $m=\rho_0\operatorname{Area}(D)$.

What is the average value of $f$ over $D$?

The average value is $f_{\mathrm{avg}}=\frac{1}{\operatorname{Area}(D)}\iint_Df(x,y)\,dA$.

How is the volume of a solid region $E$ expressed?

The volume of a solid region $E$ is $\iiint_E1\,dV$. Integrating $1$ counts volume throughout the solid.

What is the volume of $[0,2]\times[1,3]\times[-1,1]$?

The volume is $8$, since the box has side lengths $2$, $2$, and $2$, so $\iiint_B1\,dV=8$.

How is a solid described by a planar base and two surfaces?

Write $E$ as $(x,y)\in D$ with $u(x,y)\le z\le v(x,y)$, then integrate $z$ from the lower surface to the upper surface over $D$.

What is the tetrahedral volume under $z=6-2x-3y$?

The volume is $6$. The solid is in the first octant under $z=6-2x-3y$, with intercepts $3$, $2$, and $6$.