Calculus_II_application_solutions_1.nb

Solutions to Application Problems assigned in class (set 1) - Revised (I think they are correct now...)

Find the area between and (one "cell"):

In[1]:=

∫_ (-3π/4)^(π/4) (Cos[x] - Sin[x]) x

Out[1]=

Find the volume of a pyramid with an equilateral triangle base of side length 2 and height 6:

In[2]:=

∫_0^63^(1/2)/4 (y/3)^2y

Out[2]=

Find the volume of the solids of revolution:

between and , about :

In[4]:=

2π∫_ (-1)^1 (7 - x) (x^2 - x^4) x

Out[4]=

In[3]:=

π∫_0^1 ((7 - y^(1/2))^2 - (7 - y^(1/4))^2) y + π∫_0^1 ((7 + y^(1/4))^2 - (7 + y^(1/2))^2) y

Out[3]=

between and , about about (I said in the morning class and changed it to something else, which I don't remember, in the afternoon class; this formula is valid for any .

In[6]:=

π (∫_ (-2)^0 ((b - 4x)^2 - (b - x^3)^2) x + ∫_0^2 ((b - x^3)^2 - (b - 4x)^2) x)

Out[6]=

In[8]:=

2π (∫_ (-8)^0 (y/4 - y^(1/3)) (b - y) y + ∫_0^8 (y^(1/3) - y/4) (b - y) y)

between , , , about : (note: Log here means natural log here; you may approximate the integral numerically)

In[28]:=

$RowBox[{2, π, RowBox[{RowBox[{RowBox[{∫, _, 1.}], ^, 2}], (x + 3) (Log[x]) x}]}]$

Out[28]=

In[6]:=

$RowBox[{π, RowBox[{RowBox[{RowBox[{∫, _, 0.}], ^, Log[2]}], (25 - (^y + 3)^2) y}]}]$

Out[6]=

Find work:

Roll up a 40 ft chain 1/4 of the way, with a 20 lb bucket on the end (chain weighs 2 lb/ft)

In[29]:=

∫_30^402 (40 - y) y + 10 (30) (2) + 10 (20)

Out[29]=

Pump all the water out of a cone (with sharp end down) with radius 2 ft, height 8 ft

In[7]:=

$RowBox[{62.4, π, ∫_0^8 (y/4)^2 (8 - y) y}]$

Out[7]=

Pump half the water out of a sphere of radius 5 ft (assume sphere is full)

In[8]:=

$RowBox[{62.4, π, ∫_0^5 ((25 - y^2)^(1/2))^2 (5 - y) y}]$

Out[8]=

Find the fluid force ("hydrostatic force"):

[Graphics:HTMLFiles/index_36.gif]

In[9]:=

$RowBox[{9.8, (1000), ∫_0^3^(1/2) (3 - y) ((2y)/3^(1/2)) y}]$

Out[9]=

[Graphics:HTMLFiles/index_39.gif]

In[32]:=

$RowBox[{62.4, ∫_ (-3)^3 (8 - y) (9 - y^2)^(1/2) y}]$

Out[32]=

[Graphics:HTMLFiles/index_42.gif]

In[34]:=

$RowBox[{62.4, ∫_0^2 (4 - y) (2 (y/2) + 4) y}]$

Out[34]=

Created by Mathematica (March 23, 2004)