91Ó°ÊÓ

Let $f(x,y,z)$be a continuous function of three variables, let ${\mathrm{Î\copyright }}_{xy}=\left\{\right(x,y\left)\right|a≤x≤band{h}_1\left(x\right)≤y≤h_2\left(x\right)\}$be a set of points in the $xy$-plane, and let $\mathrm{Î\copyright }=\left\{\right(x,y,z\left)\right|(x,y)∈{\mathrm{Î\copyright }}_{xy}and{g}_1(x,y)≤z≤g_2(x,y)\}$be a set of points in 3-space. Find an iterated triple integral equal to the the triple integral${∭}_{\mathrm{Î\copyright }}f(x,y,z)dV$. How would your answer change if${\mathrm{Î\copyright }}_{xy}=\left\{\right(x,y\left)\right|a≤y≤band{h}_1\left(y\right)≤x≤h_2\left(y\right)\}$?

How many summands are in $\underset{i=i_0}{\overset{l}{∑}}\underset{j=j_0}{\overset{m}{∑}}\underset{k=k_0}{\overset{n}{∑}}{k}^i{j}^k$?

. To convert from rectangular to spherical coordinates

$p=\_\_\_\_,\mathrm{θ}=\_\_\_\_,\mathrm{ϕ}=\_\_\_\_$

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral ${âˆ\neg }_{\mathrm{Î\copyright }}f(x,y)dA$

${∫}_0^2{∫}_0^{-x+2}f(x,y)dydx$

Identify the quantities determined by the integral expressions in Exercises 3-11. If x and y are both measured in centimeters and $\mathrm{ÒÏ}(x,y)$is a density function in grams per square centimeter, give the units of the expression.

${âˆ\neg }_{\mathrm{Î\copyright }}{x}^2\mathrm{ÒÏ}(x,y)dA\text{and}{âˆ\neg }_{\mathrm{Î\copyright }}{y}^2\mathrm{ÒÏ}(x,y)dA$

Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.

Consider the three-petaled polar rose defined by $r=\cos 3\mathrm{θ}$. Explain why the iterated integral ${∫}_0^{2\mathrm{π}}{∫}_0^{\cos 3\mathrm{θ}}rdrd\mathrm{θ}$calculates twice the area bounded by the petals of this rose.

Let $f(x,y,z)$be a continuous function of three variables, let localid="1650352548375" ${\mathrm{Î\copyright }}_{yz}=\left\{\right(y,z\left)\right|a≤y≤band{h}_1\left(y\right)≤z≤h_2\left(y\right)\}$be a set of points in the $yz$-plane, and let localid="1650354983053" $\mathrm{Î\copyright }=\left\{\right(x,y,z\left)\right|(y,z)∈{\mathrm{Î\copyright }}_{yz}and{g}_1(y,z)≤x≤g_2(y,z)\}$be a set of points in $3$-space. Find an iterated triple integral equal to the triple integral localid="1650353288865" $\underset{\mathrm{Î\copyright }}{∭}f(x,y,z)dV$. How would your answer change iflocalid="1650352747263" ${\mathrm{Î\copyright }}_{yz}=\left\{\right(y,z\left)\right|a≤z≤band{h}_1\left(z\right)≤x≤h_2\left(z\right)\}$?