Assignment 10

Reading: 11.1, 11.2, 11.3, 11.4, and 11.7 (these are all of the sections on double and triple integrals).  Please note that in class we defined a double integral and a triple integral as

\displaystyle \iint_R \! f\, dA = \text{ limit of (signed) 3-dimensional box volumes}

\displaystyle \iiint_B \! f\, dV = \text{ limit of (signed) 4-dimensional "box" volumes}

there are other interpretations and applications of double and triple integrals.  As mentioned in our textbook, we can also interpret these integrals as telling us about average values of functions, as telling us about the total mass of an object, and even as telling us about the probability of a particular event taking place.  Please read sections 11.4 and 11.7 carefully for alternate interpretations of and uses for these integrals!  In particular, I strongly recommend reading through the summaries of these sections!

Problem 1.  Compute the volume of the solid region bounded by the graph of z = f(x,y) = ye^x and the rectangle R = [0, \ln 2] \times [-1, 1] in the xy-plane.

Problem 2.  Complete Activity 11.6 (parts a and b) from our textbook.

Problem 3.  Complete Activity 11.8 (parts a through e) from our textbook.

Problem 4.  Exercise 4(a) – 4(c) from page 207 in our textbook.

Problem 5.  Suppose f:\mathbb{R}^2\to\mathbb{R} is a given function with average value 10 over the set D = \left\{ \left(x,y\right) : y \leq x \leq 1 \text{ and } 0 \leq y \leq 2 \right\}.

(a) Draw a picture of the region D.

(b) Compute the area of D.

(c ) Compute the value of the double integral

\displaystyle \iint_D \! f\, dA.

Problem 6.  Evaluate the triple integral \displaystyle \iiint _B \! f\, dV where f(x,y,z) = xy + e^{z+y} and B = [1, 2] \times [1, 3] \times [-2, \pi].

Problem 7.  Exercises 1(a) and 1(b) from page 216.

Problem 8.  Exercise 1 from page 243.

Problem 9.  Set up a triple integral that represents the 3-dimensional volume of the solid region

\displaystyle S = \left\{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq 1 \right\}.

Note: do not evaluate this triple integral.  Although it is possible to compute, it is a bit tedious (and relies on some cool integration techniques); we will learn a better way to evaluate this integral very shortly.

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