PRACTICE TEST 3 (1710, SP013)
INSTRUCTOR: KOSHAL DAHAL
Do practice similar types problems
Q1: Express the sum in sigma notation:
a) −1 + 1 + 3 + 5 + 7
b) −1 + 2 − 3 + 4
Q2: a) Suppose the interval [1, 3] is partitioned into n = 4 subintervals. List the gridpoints, which are used for the Left, Right and Midpoint Riemann Sums.
b) Calculate the Left and Right Riemann sum for f (x) = Cosx on [0, π2 ] with n = 4.
Q3: Find the net-area geometrically of y = 2x + 4 on [−4, 2]
Q4: Evaluate the Indefinite integral
a)
R
8x Cos(4x2 + 3) dx
R
b) (x6 − 3x2 )34 (x5 − x) dx
c)
√
( x+1)4
√
2 x
R
dx
Q5: Evaluate the Definite integral
a)
Rπ
0
R9
b) 1
(1 + Sin7x)2 Cos7x dx
√
8− x
√ dx
x
R2
3
c) −2 (x9 − 3x5 + 2x2 + 10Sinx + π (x(x2−4x)
) dx
+1)
d)
R π/2
Cosθ
π/4 Sinθ
dθ
1
2
INSTRUCTOR: KOSHAL DAHAL
Q6: Find the derivatives
a)
d
dx
Rx
b)
d
dx
R x2
c)
d
dx
R cosx
d)
d
dx
1
o
o
sin2 t dt
cos t2 dt
(t4 + 6) dt
R tanx √
o
t dt
Q7: Suppose a moving object has velocity V (t) = 6 + 2t on [0, 8], with S(0) = 0.
Then find
(a): Distance traveled
(b): Position function
Q8: A 200 − L Cistern√is empty when water begins flowing into it (at t=0) at a rate
L/min given by Q0 (t) = 3 t. Note, Q(0) = 0. Then find when the tank will be full?
Q9: Find the area (both w.r.to x and w.r.to y) bounded by: y = 4 − x2 and y = x + 2
√
Q10: Find the volume generated when R is revolved about the x-axis, for y = 2 x, y = x
Q11: Use√Shell method to find the volume generated when R is revolved about the y-axis,
for g(x) = x, & f (x) = x2
Q12: A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where
0 ≤ h ≤ 8 (h=0 corresponds to an empty bowl). Use either the Shell method or the
Washer method to find the volume of water in the bowl as a function of h.
Q13: Find the arc length w.r.to x, for y =
x6
24
−
5
3x2
on [1, 3].
PRACTICE TEST 3 (1710, SP013)
3
Q14: A Hot Spring has the shape of a box with a base that measures 33m by 22m and a
depth of 4.5m. How much work is required to pump the water out of the Hot Springs when
it is full?
Q15: The profile of the cable on a suspension bridge may be modeled by a parabola
y = 0.00037x2 (gives a good fit to the shape of the cables, where |x| ≤ 640 and x & y are
measured in meters). The central span of the Golden Gate Bridge is 1280m long and the
152m high. Then approximate the length of the cables that stretch between the tops of
the two towers.
Good Luck!