Math A1a (Calculus)
Riemann sum and integration
Dr. Mohamed Khaled
P
√
1. For f (x) = x on the interval [3, 4], calculate the Riemann sum 3j=1 f (sj )∆x where the
sample points s1 , s2 , s3 are given according to:
(a) the left endpoint rule
(b) the right endpoint rule
(c) the trapezoidal rule
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2. Estimate the Riemann integral 1 x1 dx by the Riemann sum that arises from 4 subintervals
and the choice of sample points s1 , s2 , s3 , s4 in which each sample point sj is a distance 13 ∆x
from the left endpoint xj1 of the subinterval [xj1 , xj ] that contains it.
3. Let f (x) = (x − 1)(x − 2)(x − 4), where 1 ≤ x ≤ 4. The area of the region under the graph
5
of f and above the interval [1, 2] of the x-axis is 12
. The area of the region above the graph
8
of f and under the interval [2, 4] of the x-axis is 3 . Use the given information to evaluate the
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definite integral 1 f (x) dx.
Notice that the verical axis shown is the line x = 1, not the y-axis.
4. Let f (x) = (x − 1)(x − 2)(x − 4), where 1 ≤ x ≤ 4.
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(a) Estimate 1 f (x) dx using the midpoint rule with n = 9 in Riemann’s sum.
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(b) Estimate 1 f (x) dx using the maximum rule with n = 9 in Riemann’s sum.
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(c) Estimate 1 f (x) dx using the minimum rule with n = 9 in Riemann’s sum.
5. Consider an isosceles right triangle such that both the twin sides are of length 1. One of
the twin sides is placed horizontally such that the other twin meets its left endpoint. The
horizontal side is divided into n-many equal segments. On each of these segments, a rectangle,
of height equal to the distance between the left endpoint of segment and the hypotenuse, is
placed. Let Sn be the sum of the areas of all of those rectangles. Show that limn→∞ Sn is
equal to the area of the triangle.
6. Use the definition of definite integral (Riemann Sum) to evaluate the following integrals:
Z
6
2
Z
2
2x dx
4
3
Z
x dx
−1
1
1
5
(x − 4x2 ) dx
Math A1a (Calculus)
Riemann sum and integration
Dr. Mohamed Khaled
7. Use the definition of definite integral (Riemann Sum) to evaluate the following integrals:
Z
5
Z
3
Z
2
−1
(5x − 2x) dx
4x dx
0
(x2 + 3x + 5) dx.
−5
0
8. (a) By finding a partition of the interval [0, 1], using the function f (x) = ex and a suitable
Riemann sum show that
n−1
1
2
1
(1 + e n + e n + · · · + e n ) = e − 1.
n→∞ n
lim
1
2
(b) Find this limit again by explicitly finding the sum (1 + e n + e n + · · · + e
L’Hospital’s rule.
9. Show that
lim
n→∞
n−1
n
1
1
(1k + 2k + 3k + · · · + nk ) =
.
k+1
nk+1
10. Show that
√
1
1
1
1
1
+√
+√
+ · · · + √ ) = 2( 2 − 1).
lim √ ( √
n→∞
n 1+n
2+n
3+n
2n
11. (a) Calculate F 0 (x) and F 0 (3) for F (x) =
(b) Calculate G0 (x) and G( π6 ) for G(x) =
19+t4
0 1+t2
R 2π √
17
x
Rx
12. Evaluate the following integrals:
R1
(a) −1 x3 dx.
R0
(b) −3 (2x + 6)4 dx.
Rπ
(c) −2 π cos x dx.
2
R2 t
(d) ln 1 e − e−t dt.
Rπ2
(e) −π sin x dx.
2
dt.
+ 16 sin s ds.
) and using