University of Namibia
Department of Computing, Mathematical and Statistical Sciences
MAT 3612 Calculus 2 Tutorial 3
August 15, 2023
1. In the following you may use the Fundamental Theorem of Calculus part
1.
a) Find a function f and a number a such that
Z x
√
f (s)
ds = 2 3 x for all x > 0.
10 +
3
s
a
b) If f is a continuous function such that
Z x
Z
f (r)dr = xe3x +
0
x
e−t f (t)dt
0
for al x ∈ R, find an explicit formula for f (x).
R x+√3 √
c) Find lim x1 √3
1 + t2 dt.
x→0
d) If x sin(2πx2 ) =
f (27).
R x3
0
f (t) dt, where f is a continuous function, find
e) If f is a differentiable function such that f (x) is never 0 and
Z x
f (t) dt = [f (x)]2
1
for all x ∈ R , find the explicit formula for f (x).
R x2
f ) If f (x) = 0 x3 sin(t2 ) dt, find f 0 (x).
R x R cos t √
d2
1 + s4 ds dt
g) Find dx
2
0
1
2. Consider the function f (x) = ln(x + 1).
1
a) Use six subintervals to find estimates of each type for the area under
the given graph of f from x = 0 to x = 12.
(i) L6 (Left point Rule)
(ii) R6 (Right point Rule)
(iii) M6 (Midpoint Rule)
(iv) T6 (Trapezium Rule)
b) Find the true area under the graph of f from x = 0 to x = 12. (i.e
R 12
find 0 ln(x + 1) dx.)
c) Which of the approximations you found in a) is an underestimate of
the true area?
d) Which of the approximations you found in a) is an overestimate of the
true area?
e) Which of the numbers L6 , R6 , M6 or T6 gives the best estimate? Explain.
2
3. Consider f (x) = e−x for −2 ≤ x ≤ 2.
a) Sketch the graph of f .
b) Estimate the area under the graph of f using 8 approximating rectangles and taking the sample points to be midpoints. Sketch the curve
and the rectangles.
c) Estimate the area under the graph of f using 8 approximating trapeziums. Sketch the curve and the trapeziums.
d) By using your answer in b) and c), which of the methods gives the best
estimate? Explain.
4. If f (x) = ex − 2 for 0 ≤ x ≤ 4.
a) By using Trapezium rule, evaluate
R4
0
f (x)dx.
b) Evaluate the Riemann sum for f from x = 0 to x = 4 by using Right
point rule with six subintervals. Does your answer represents an over
estimate or an under estimate?
5. Use the Trapezium Rule with the given value of n to approximate the
integral. Round the answer to four decimal places. In each case state the
error bound ET .
R 10 √
a) 2
x3 + 1dx, n = 4
R1
b) 0 sin(x2 )dx, n = 5
R1
c) 0 cos4 (x2 )dx, n = 6
R5
d) 1 x2 e−x dx, n = 4
6. By using Riemann sums, prove that
2
a)
b)
Rb
a
Rb
a
b2 −a2
2
xdx =
x2 dx =
b3 −a3
3
7. Determine a function f and an interval [a, b] such that the area under the
graph of f from x = a to x = b is equal to the given limit. Do not evaluate
the limit.
n
X
2k 10
2
5+
a) lim
n→∞
n
n
b) lim
n→∞
k=1
n
X
k=1
πk π
tan
4n
4n
8. Express the limit as a definite integral on the given interval.
a) lim
n→∞
b) lim
n→∞
n
X
k=1
n
X
k=1
xk ln(1 + (xk )2 )∆x,
cos xk
∆x,
xk
[2, 6]
[π, 2π]
9. Express the integral as a limit of Riemann sums. Do not evaluate the
limit.
R7 x
a) 2 1+x
6 dx
R 10
b) 0 (x − 2) ln(x + 1)dx
10. Evaluate the limit by first recognizing the sum as a Riemann sum for a
function defined on [0, 1].
q
q
q
pn
1
2
3
a) lim n1
n +
n +
n + ...... +
n .
n→∞
n
1 X k 2
.
n→∞ n
n
b) lim
k=1
11. Show that cos(x2 ) ≥ cos x for 0 ≤ x ≤ 1. Hence use comparison properties
R π/6
of definite integrals to deduce that 0 cos(x2 ) dx ≥ 21 .
3