91影视

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

${鈭�}_{惟}\left(x^2+y^2\right)蚁(x,y)dA$

Explain how the symmetries of the graphs of polar functions can be used to simplify area calculations.

Evaluating iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.

${鈭�}_0^2{鈭�}_0^{y}x\sqrt{y^2-x^2}dxdy$

Explain how to construct a Riemann sum for a function of two variables over a rectangular region.

To convert from cylindrical to spherical coordinates:

$蚁=----,胃=----,蠒=----$

Which of the iterated integrals in Exercises 9鈥�12 could correctly be used to evaluate the double integral${鈭�}_{惟}f(x,y)dA,$

${鈭�}_0^2{鈭�}_0^{-y+2}f(x,y)dxdy$

Let $蚁(x,y,z)$be a density function defined on the rectangular solid $R$where role="math" localid="1650358913892" $R=\left\{\right(x,y,z\left)\right|鈭�1鈮�x鈮�3,0鈮�y鈮�2,and2鈮�z鈮�7\}$. Set up iterated integrals representing the mass of $R$, using all six distinct orders of integration.

Explain how to construct a midpoint Riemann sum for a function of two variables over a rectangular region for which each $\left(x_j^{*},y_k^{*}\right)$ is the midpoint of the subrectangle

Refer to your answer to Exercise 10 or to Definition 13.3.

Give a geometric explanation why $n{鈭�}_0^{2蟺/n}{鈭�}_0^{R}rdrd胃=蟺R^2$

for any positive real number $\mathrm{鈩�}$and any positive integer $n$.

Would the equation also hold for nonintegral values of $n$?

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

role="math" localid="1650627371293" $\sqrt{\frac{{鈭�}_{\mathrm{惟}}鈥�{\mathrm{x}}^2\mathrm{蚁}(\mathrm{x},\mathrm{y})\mathrm{dA}}{{鈭�}_{\mathrm{惟}}鈥�\mathrm{蚁}(\mathrm{x},\mathrm{y})\mathrm{dA}}}$androle="math" localid="1650627382876" $\sqrt{\frac{{鈭�}_{\mathrm{惟}}鈥�{\mathrm{y}}^2\mathrm{蚁}(\mathrm{x},\mathrm{y})\mathrm{dA}}{{鈭�}_{\mathrm{惟}}鈥�\mathrm{蚁}(\mathrm{x},\mathrm{y})\mathrm{dA}}}$