Math 52: Homework 1
Due Tuesday April 11
April 6, 2017
You are encouraged to work with your classmates to discuss these problems,
but you must write your own solutions individually and in your own words.
You must show your work and use sentences and pictures for justification.
1. (5pts) Estimate the volume of the sand in 3 × 4m rectangular sandbox in whose four
corners the depth of the sand is 30 cm, 60 cm, 70 cm and 100 cm. Explain the choices
you made for your estimate.
2. (5pts) The double integral
ZZ
q
R2 − x2 − y 2 dA
x2 +y 2 ≤R2
is the volume of a familiar solid. Describe that solid. Use that knowledge (rather than
integration) to determine the value of the integral.
3. (10pts) Calculate the Riemann sum for the function f (x, y) = x2 y 2 over the region
R = [1, 3] × [2, 5] partitioned into six 1 × 1 squares using the function value in the
lower left corners. Think about whether the Riemann sum using the function value in
the upper right corner will be larger or smaller than the lower left estimate. Explain
with sentences and/or pictures why it will be larger or smaller. (You do not need to
calculate the Riemann sum using the upper right corner, and your explanation should
not be this calculation, but you can do the calculation on your own to check if your
reasoning is correct.)
4. (8pts) For each of the following: Evaluate the integrals. Draw the regions in the xyplane which you are integrating over in each example. Describe each of these regions
with inequalities.
(a)
Z 1Z 2
0
ey xdxdy
0
(b)
Z 1Z 2
0
1
0
ey xdydx
5. (12pts) For each of the following, determine if the equality is always true or give an
example where the equality is not true (counterexample).
For f (x, y) a continuous function f (x, y) and a a positive number
(a)
Z aZ 1
2 · f (x, y)dxdy = 2
Z aZ 1
0
0
f (x, y)dxdy
0
0
(b)
Z 2a Z 1
f (x, y)dxdy = 2
Z aZ 1
0
0
f (x, y)dxdy
0
0
(c)
Z 1Z a
0
f (x, y)dxdy =
0
Z aZ 1
0
f (x, y)dxdy
0
(d)
Z 1Z a
0
f (x, y)dxdy =
Z aZ 1
0
0
f (x, y)dydx
0
6. (4pts) Let R be the rectangle with −1
≤ x ≤ 1 and −2 ≤ y ≤ 2. Which symmetries
R
of a function f (x, y) guarantee that R f (x, y)dA = 0? Choose all that apply.
(a) f (−x, y) = f (x, y)
(b) f (−x, y) = −f (x, y)
(c) f (x, −y) = −f (x, y)
(d) f (−x, −y) = −f (x, y).
7. (6pts) Let M be the solid bounded by the graph of f (x, y) = x2 y 3 , the xy-plane, and
the planes x = −2, x = 1, y = −1 and y = 1.
(a) Explain why the volume of M is not equal to
Z 1 Z 1
−2
x2 y 3 dydx
−1
(b) Find the volume of M .
8. (5pts) Interpret the value of the integral:
Z 2 Z 2−x
0
(x + 2y)dydx
0
as the volume of some solid. Compute that volume using the geometry and the fact that
1
the volume of the pyramid spanned by the vectors ~v , w
~ and p~ is equal to | det [~v , w,
~ p~ ]|.
6
2
9. (6pts) Sketch the following regions:
(a) 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 − x2
(b) x2 + y 2 ≤ 4 and y ≥ 1
(c) x ≤ y 2 and y ≥ x3
10. (18pts) Sketch the region and evaluate the integral:
(a)
Z 2 Z 2x
(1 − y)dydx
0
0
(b)
Z 1Z x
x4
0
(y − x)dydx
(c)
Z π Z sin(x)
ydydx
0
0
(d)
Z π Z x3
0
ey/x dydx
0
11. (21pts) Sketch the region, switch the order of integration, and then calculate the integral.
(a)
Z 1Z 1
0
2
e−x dxdy
y
(b)
Z πZ π
0
(c)
x
sin(y)
dydx
y
Z √π Z √π
0
y
3
sin(x2 )dxdy