Dplatte
Calculus III
Dr. Platte
Test II
March 18, 2014
Name:_____________________________________________________
√9 − 𝑥 2 − 𝑦 2 𝑑𝐴. Where R is the
1.
(14 points) Use a double integral in polar coordinates to find value of ∬
𝑅
region in the first quadrant within the circle 𝑥 2 + 𝑦 2 = 9.
2.
(14 points) Express the integral as an equivalent integral with order of integration reversed.
4
8
∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦
0
2𝑦
3.
(14 points) Use the chain rule to find
z
z
, where 𝑧 = 𝑥 2 𝑦 − 2𝑥 + 𝑦 𝑎𝑛𝑑 𝑥 = 𝑢𝑣, 𝑦 = 𝑢 − 𝑣 . Leave
and
u
v
answer in mixed variable form.
4.
(14 points) Use the total differential (or local linear approximation) to approximate the value of 𝑓(𝑥, 𝑦) = √1 + 𝑥𝑦
at point Q(3.99,2.01), knowing the value of f( x, y) at point P (4,2).
5.
(14 points) Use the formula (method for implicit differentiation) developed in this course to find dy/dx for 𝑦 3 +
2𝑥 2 𝑦 = 3 .
6.
(15 points) Find volume of the solid bounded above by z = 1 – y2 , on the bottom by z = 0 (xy plane) and on the sides
by x = 0 (yz plane), y = 0 (xz plane) and the plane y = -x +1.
7.
(15 points) Evaluate the triple integral ∭
𝐺
parabolic cylinder y = 1 – x2.
1𝑑𝑉 . Where G is the solid enclosed by plane z=y, the xy plane, and the