I didn't believe the premises on which he based his argument, so he clearly didn't convince me of his conclusion.

Describe the steps you would follow to find the limits of integration that express a double integral over a nonrectangular region as an iterated double integral. Illustrate your discussion with an example.

Find the area of the region between the curves $y=2 x^2+x$ and $y=x^2+2$ from x=0 to x=2.

Explain the "duet" and "octet" rules and how they are used to describe the arrangement of electrons in a molecule.

Sketch a direction field for the differential equation. Then use it to sketch three solution curves.

$$y^{\prime}=x y+y^2$$

Change the indicated order of integration to each of the other five orders. $\int_{0}^{2} \int_{0}^{4-2 y} \int_{x+2 y}^{4} F(x, y, z) d z\ d x\ d y$

Evaluate each definite integral. Use the integration feature of a graphing calculator, if you wish, to support your answers. $\int_{\pi / 6}^{\pi / 4} \sin x d x$

The following table gives the average hourly earnings (in dollars) of U.S. production workers from 1970 to 2010.

If you have appropriate technology, verify that the least-squares regression line that models these data (with coefficients rounded) is y = .343 x-20.65, where y is the average hourly wage in year x and x = 0 corresponds to 1900.

Use the rules of exponents to simplify the expression.

$$-5\left(y^{4}\right)^{3}$$

Each argument either is valid by one of the forms of valid arguments discussed in this section, or is a fallacy by one of the forms of invalid arguments discussed. Determine whether the argument is valid or a fallacy, and give the form that applies.

"If you're going through hell, keep going." (quote from Winston Churchill).

$$\underline{\textnormal{You're not going through hell.}\qquad\quad}$$

Don't keep going.

One number added to twice another number is $13$. Four times the first number added to twice the other number is $-2$. What are the numbers?

Find the area of the region bounded by the given curves.

$$y=e^x-1, \quad y=x^2-x, \quad x=1$$

If 4 is added to twice a number and this sum is multiplied by 2, the result is the same as if the number is multiplied by 3 and 4 is added to the product. What is the number?

Compare the curves represented by the parametric equations $x = e^t$ and $y = e^{-2t}$. How do they differ?