Problem 1
When integrating \(f_{x}(x, y)\) with respect to \(x\), the constant of integration \(C\) is really which: \(C(x)\) or \(C(y) ?\) What does this mean?
Problem 1
An integral can be interpreted as giving the signed area over an interval; a double integral can be interpreted as giving the signed ________ over a region.
Problem 1
Why is it easy to use "mass" and "weight" interchangeably, even though they are different measures?
Problem 1
Explain the difference between the roles \(r,\) in cylindrical coordinates, and \(\rho\), in spherical coordinates, play in determining the location of a point.
Problem 2
Integrating an integral is called ______ ______ .
Problem 2
Explain why the following statement is false: "Fubini's Theorem states that \(\int_{a}^{b} \int_{g_{1}(x)}^{g_{2}(x)} f(x, y) d y d x\) \(\int_{a}^{b} \int_{g_{1}(y)}^{g_{2}(y)} f(x, y) d x d y . "\)
Problem 2
Why would one be interested in evaluating a double integral with polar coordinates?
Problem 2
Why are points on the \(z\) -axis not determined uniquely when using cylindrical and spherical coordinates?
Problem 2
Give an informal interpretation of what \(" \iiint_{D} d V^{\prime \prime}\) means.
Problem 3
Explain why if \(f(x, y)>0\) over a region \(R,\) then \(\iint_{R} f(x, y) d A>0\).
