Calculus

1 . Consider the region bounded by the graphs of y = sec at, y = 0,:1: = -7r/4,and :1: = 7r / 4. Find
Mm area of this region and determine the surface area and volume of the solid formed by
1 Of 2 1mrolving this about the x-axis.
z. bind the indefinite integral of
(tan:r + sec :13)2d:1:, and /sec2 :1:(1 + tan2 x)-1d:1:

3. Consider the region bounded by the graphs of a: = y3, and :1: ∶ 2y – y2. Find the area of this
region and determine the surface area and volume of the solid formed by revolving this about
the y-axis.

4. A palm tree is growing at the rate of % ft. / yr., where t is in years for 0 ≤ t ≤ 12.

a What is the tree’s rowth rate when t = 0 ears?
g y
b W hat is the tree’s rowth rate when t = 9 ears?
g y
c W hat is the hei ht of the tree at the end of the 12-year rowth period?
9; . g
5. You go on a bike ride. Assume your velocity at time t(hours) is t\/ 1‘? + 3miles/hour for
0 ≤ t ≤ 5.
(a) What is your velocity when t : 1 hour?
(b) What is your velocity when t = 3 hours?
(c) What is the total distance traveled for 0 ≤ t ≤ 5?

6. Water is leaking from a large tank at the rate of (1 / 4)t2 gal. / hr., where t is given in hours

for t ≥ 0.
(a) What is the leakage rate when t ∶ 1 hour?
(b) What is the leakage rate when t = 5 hours?
(c) W hat is the total amount of water which leaks from the tank for 0 ≤ t ≤ 6?
d If the tank initially holds 100 allons, how 1011 will it take for the tank to become
9; g
empty?

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7. A thin wire lies along the x-axis from :1: = 0 to :1: = 271cm. It’s density at :1: is given to be

3 + sin 21: grams / cm.

(a) What is the wire’s density at :1: = 0 cm.?
(b) What is the wire’s density at :1: = 11/4cm.?
(c) What is the wire’s density at :1: = 771 /4cm.?
((1) What is the total mass of the wire?
(e) What is the mass of the wire from :1: = 7r / 2cm. to :1: ∶ 1r.cm?
8. Find the area bounded by the graphs of
(a) m:y2 andx=9
(b) 33:11:? andm=y+2
(C) x=y(y-2) and :v=y(1-2y).