91Ó°ÊÓ

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.

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral ${âˆ\neg }_{0}f(x,y)dA$

${∫}_2^0{∫}_{-y+2}^{0}f(x,y)dxdy$

To convert from spherical to cylindrical coordinates:

$r=----,\mathrm{θ}=----,z=----$

What is the difference between a double integral and an iterated integral?

Complete Example 2 by showing that

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{1}{4}\mathrm{π}+\frac{9}{16}\sqrt{3}$

and

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{3}{8}\mathrm{π}-\frac{9}{16}\sqrt{3}$

Let $\mathrm{ÒÏ}(x,y,z)$be a density function defined on the rectangular solid $R$where role="math" localid="1650360487967" $R=\left\{\right(x,y,z\left)\right|a_1≤x≤a_2,b_1≤y≤b_2,and{c}_1≤z≤c_2\}$. Set up iterated integrals representing the mass of $R$, using all six distinct orders of integration.

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.

${∫}_0^{\sqrt{\mathrm{π}}}{∫}_x^{\sqrt{\mathrm{π}}}\sin y^{2}dydx$

Throughout this section we computed several integrals relating to the triangular region $\mathrm{Î\copyright }$with vertices (1, 1), (2, 0), and (2, 3). In Exercises , you are asked to provide the details of those computations.

Show that the area of $\mathrm{Î\copyright }$is $\frac{3}{2}$by using the area formula for triangles and by evaluating the integral role="math" localid="1650628428282" ${∫}_1^2 {∫}_{−\mathrm{x}+2}^{2\mathrm{x}−1} \mathrm{dydx}$

Following region is bounded by functions $y=\frac{1}{2}x\text{and}y=\sqrt{x}$.

Express $\mathrm{Î\copyright }$as type I or type II region. If $\mathrm{Î\copyright }$is a type I region, what are

$a,b$? If $\mathrm{Î\copyright }$is a type II region, what are $c,d$?

What are the six forms used to express the volume increment $dV$when you use rectangular coordinates to evaluate a triple integral? How do you decide which order to use?