Calculus Answers

Let D={(x,y,z)∈R^3:1≤x≤3,0≤y≤ln(z),0≤z≤x}. The answer to ∭D e^yln(x)dV

is
Select one:
a. 13/9
b. 3/2
c. 5/9
d. -13/9
e. 2
f. 1/2

Let D={(x,y,z)∈R^3:−1≤x≤2,0≤y≤1,0≤z≤π}. The answer to ∭D yzdV

is
Select one:
a. (3π)/4

b. 3π^2

c. 3/4

d. 75π^2


e. (3π^2)/4

The density of a semicircular plate with radius a>0 is described by ρ(x,y)=√(x^2+y^2).

The mass of the plate is
Select one:
a. (πa^3)/3


b. πa^2

c. a^3/3


d. (πa^2)/2

Let D={(x,y)∈R^2:y^2<x<y and 0<y<1}. The answer to ∬D 2y^2xdA

is
Select one:
a. 70
b. 1/35
c. -70
d. 2/35
e. 35

Compute ∬D(x+y)dA where D={(x,y)∈R^2:0≤x≤2 and 0≤y≤√(4-x^2)}

.
Select one:
a. 16
b. 8
c. 8/3
d. 16/3
e. 3

Let f(x,y)=y/x. Evaluating ∫f(x,y)dx

leads to


Select one:
a. −yln|x|)+g(x)
where g(x) is a function of x
.

b. yln|x|+g(y)
where g(y) is a function of y
.
c. ln(xy)


d. ln|y|+g(y)
where g(y) is a function of y
.
e. ln(xy)+g(y)
where g(y) is a function of y
.

f. ln(x)

Let D={(x,y,z)∈R^3:x^2+y^2≤4 and 0≤z≤5}. The answer to ∭D zdV is: Select one:
a. 50
b. 3
c. 50π

d. 2π

e. π^2

f. 2

The lifetime of a machine is continuous on the interval (0, 40) with probability density
function f, where f(t) is proportional to (t+ 10)−2
, and t is the lifetime in years. Find
the proportionality constant that makes the f(t) a probability function.

Differentiate each function given below with respect to x.

(i) f(x) = √tan^−1x

(ii) f(x) = cos^−1(e^2x)

(iii) f(x) = sin^−1(√1− √x^2 )

(iv)f(x) = tan^−1(x/√x-x^2)

(v) f(x) = sin( tan^−1 4x)

f(x) = √secx.x³