91Ó°ÊÓ

Q. 7${∫}_0^1{∫}_{-\sqrt{1-y^2}}^{\sqrt{1+y^2}}\frac{x}{y+1}dxdy$

Problem Zero: Read the section and make your own summary of the material.

Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let $R=\left\{(x,y)|0≤x≤2and1≤y≤4\right\}$.

$âˆ\neg (x+2y)dA$

Let $[a_1,a_2],[b_1,b_2],[c_1,c_2]$be three closed intervals explain why the triple integral ${∫}_{a_1}^{a_2}{∫}_{b_1}^{b_2}{∫}_{c_1}^{c_2}dzdydx$computes the volume of the rectangular solid with length$a_2-a_1$, width$b_2-b_1$ and height$c_2-c_1$

Let be real $\mathrm{Î\pm },\mathrm{β},\mathrm{γ},\mathrm{δ},\mathrm{Ï\mu },\text{and}\mathrm{Î\P }$numbers. Evaluate the triple iterated integral

${∫}_{\mathrm{Î\pm }}^{\mathrm{β}}{∫}_{\mathrm{γ}}^{\mathrm{δ}}{∫}_{\mathrm{Ï\mu }}^{\mathrm{Î\P }}dzdydx$

What does this integral represent?

Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral

$\frac{1}{2}{∫}_0^{\mathrm{π}}{\cos }^{2}3\mathrm{θ}d\mathrm{θ}$

$\frac{{âˆ\neg }_{\mathrm{Î\copyright }}xdA}{{âˆ\neg }_{\mathrm{Î\copyright }}dA}\text{and}\frac{{âˆ\neg }_{\mathrm{Î\copyright }}ydA}{{âˆ\neg }_{\mathrm{Î\copyright }}dA}$

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

${∫}_0^{-x+2}{∫}_0^{2}f(x,y)dxdy$

To convert from spherical to rectangular coordinates:

$x=-----,y=-----,z=-----$

Let $f(x,y,z)$be a continuous function of three variables, let role="math" localid="1650355225242" ${\mathrm{Î\copyright }}_{xz}=\left\{\right(x,z\left)\right|a≤x≤band{h}_1\left(x\right)≤z≤h_2\left(x\right)\}$be a set of points in the $xz$-plane, and let $\mathrm{Î\copyright }=\left\{\right(x,y,z\left)\right|(x,z)∈{\mathrm{Î\copyright }}_{xz}and{g}_1(x,z)≤y≤g_2(x,z)\}$be a set of points in 3-space. Find an iterated triple integral equal to the triple integral $\underset{\mathrm{Î\copyright }}{∭}f\left(x,y,z\right)dV$. How would your answer change if${\mathrm{Î\copyright }}_{xz}=\left\{\right(x,z\left)\right|a≤z≤band{h}_1\left(z\right)≤x≤h_2\left(z\right)\}$?