Math 2205 - SERCalc II
Quiz 4
Name:
1.(3 points)(a) Derive a formula for the area of an equilaterial triangle of side length s.
(b) Use horizontal slicing to find the volume of a right equilaterial pyramid with base side-length 2 ft and height
5 ft. (Hint: Horizontal slices are all equilaterial triangles and the side-length of these slices varies linearly with
height. You know that s(0)=2 and s(5)=0)
2.(3 points) A 10 foot tall inverted conical tank of radius 5 ft at the top is buried with its top 2 feet below the
ground (the tank in an inverted right circular cone of height 10 ft and base radius 5ft. Its top is 2 ft below ground
level). If the tank is initially filled to half its height with water (64 lbs/ft3 ), find the work required to pump all
of the water in the tank to ground level. Write your final answer in the form a π where a is an integer and include
units.
3.(2 points) Sketch graphs of f (x) = ln(x) and g(x) = x ln(x) for x ∈ Z
(0, e2 ]. Pay attention to both scale and
e
x ln(x) dx. Finally, show all details in
relative shape. Next, indicate the the signed area being calculated by
the calculation of the exact value of this definite integral.
1
e
4.(2 points) (a) Use the limit definition of the derivative to prove
d 3
x = 3x2 .
dx
1
d 1
= − 2.
(b) Use the limit definition of the derivative to prove
dx x
x