assignment

Name:

• I will be checking for organization, conceptual understanding, and proper mathematical communication, as well as completion of the problems.

• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing.

• Use correct mathematical notation. • You may work with your classmates. However, please submit your own work!

1. Consider a triangular region T bounded by (i) y = x, (ii) y = 2x, and (iii) x = 2. Let f(x, y) = y

x2 + y2 .

(a) (2 points) Sketch and shade the region T described.

(b) (8 points) Set up an integral for

∫∫ T f(x, y) dy dx, then evaluate. Do all integrations without

a calculator. Obtain the EXACT VALUE for the integral!

(BONUS - 2 points) Set up an integral for

∫∫ T f(x, y) dx dy, then evaluate. Do all integrations

without a calculator. Obtain the EXACT VALUE for the integral! (Hint: The following integral formula might be helpful!∫ b

a

1

x2 + k2 dx =

1

k arctan

(x k

)∣∣∣∣b a

for any constant k.

Also, one of the resulting integrals needs to be integrated by parts. FYI: This is an EXTREMELY TEDIOUS problem... Do expect at least 2∼3 pages worth of calculations!)

2. (10 points) Average value Recall from Calculus II that the average value of a single-variable function over an interval [a, b] is given by

1

b−a

∫ b a

f(x) dx

.

(1)

Let f(x) =

∫ 1 x

et 2

dt. Use Equation (1) above to determine the EXACT average value of f on the

interval [0, 1]. Do all integrations without a calculator. Your answer should NOT involve any integrals!

1