Calculus 2: Worksheet 3 MATH 1700

4
Intended Learning Outcomes
Z
(b) ( x2 − 4 x + 2) dx
−4
• Express the limit of a Riemann sum as a definite Z 0
integral on a given interval. (c) ( x2 + x) dx
−2
Z 2
• Evaluate definite integrals using the limit defini- (d) (2 x − x3 ) dx
tion and known summation formulas. 0
Z 1
• Interpret the value of a definite integral as the (e) ( x3 − 3 x2 ) dx
0
signed area under a curve.
3. (a) Each of the regions G, H, and I bounded
• Apply the Fundamental Theorem of Calculus to by the graph of f and the x–axis
differentiate functions defined by integrals. has area 2, 1, and 3, respectively.

• Compute definite integrals of algebraic and trigono-

metric functions using antiderivatives. y

G I

x
Exercises -2 0 1 4
H
1. Express the limit as a definite integral on the given
interval. Find the value of
Z 4
n 1 − x2 £ ¤
j f ( x) + 3 dx.
∆ x,
X
(a) lim 2
[2, 6]. −2
n→∞
j =1 4 + x j
n cos x
(b) Each of the regions D, E, and F bounded by
j
∆ x, the graph of f and the x–axis has area 4.
X
(b) lim [π, 2π].
n→∞
j =1 xj
y
n £
5 ( x∗j )3 − 4 x∗j ∆ x,
X ¤
(c) lim [2, 7].
n→∞
j =1 D F
n x∗j
∆ x,
X
(d) lim ∗ 2 [1, 3]. x
n→∞
j =1 ( x j ) + 4
-3 -1 1 3
E
2. Use the definition of definite integral to evaluate
the integrals below. You may use the following Find the value of
equalities: Z 3
n n( n + 1)
£ ¤
X f ( x) + 2 dx.
i= −3
i =1 2
(c) Each of the regions A , B, and C bounded
n n( n + 1)(2 n + 1) by the graph of f and the x-axis has area 3.
i2 =
X
i =1 6 y

n n( n + 1)
· ¸2 B
3
X
i =
i =1 2
-4 -2 0 2 x
Z 5
A C
(a) (4 − 2 x) dx
2

Dr. Angel Barria Comicheo

Calculus 2: Worksheet 3 MATH 1700

Z 1
Find the value of
(c) ( u + 2)( u − 3) du
Z 2 0
£ ¤
f ( x) − 5 dx.
Z 4 p
−4 (d) (4 − t) t dt
0
4. Find the derivative of the function.
Z 9 x−1
(e) p dx
Z x
1 1 x
(a) g( x) = 3
dt. Z 2
1 t +1
Z x (f) ( y − 1)(2 y + 1) d y
¢5 0
2 + t4 dt.
¡
(b) g( x) = Z π/4
1
Z s
¢8 (g) sec2 t dt
t − t2 dt. 0
¡
(c) g( s) =
5
Z π/4
Z rp (h) sec θ tan θ d θ
(d) g( r ) = x2 + 4 dx. 0
0 Z 2
(1 + 2 y)2 d y
Z πp
(i)
(e) F ( x) = 1 + sec t dt. 1
x Z 2 s4 + 1
Z 1 (j) ds
s2
¡p ¢
(f) G ( x) = cos t dt. 1
x
v5 + 3 v6
2
Z
Z 1/x (k) dv
(g) h( x) = sin4 t dt. 1 v4
2 s
p Z 18
Z
z2 x 3
(h) h( x) = dz. (l) dz
z4 + 1 1 z
1 Z π
Z tan x q
p (m) f ( x) dx, where
(i) y = t + t dt. 0
0
x 4
π
Z 
2 sin x, 0≤x< ,
(j) y = cos θ d θ .

0 f ( x) = π 2
1 3 cos x,
 ≤ x ≤ π.
u
Z
2
(k) y = 2
du.
1−3x 1 + u
Z 1 p Z 2

(l) y = 1 + t2 dt. (n) f ( x) dx, where

−2
sin x
3x u2 − 1
Z 
(m) g( x) = du.
2, −2 ≤ x ≤ 0,
2x u2 + 1 f ( x) = 2
Z 1+2x 4 − x , 0 < x ≤ 2.
(n) g( x) = t sin t dt.
1−2x Z 9 1
Z x 3 (o) dx
(o) h( x) = cos t2 dt.
¡ ¢
1 2x
p
x
Z 1
x 2 (p) 10 x dx
1
Z
0
(p) g( x) = p dt. p
tan x 2 + t4 3
6
Z
2
(q) 1
p dt
5. Evaluate the integral. 2 1 − t2
Z π Z 1
4
(a) sin θ d θ (r) dt
π/6 0 t2 + 1
Z 5 Z 1
(b) π dx (s) e u+1 du
−5 −1

Dr. Angel Barria Comicheo

Calculus 2: Worksheet 3 MATH 1700

4 + u22
Z
(t) du
1 u3
Z 4
p
(u) t (1 + t) dt
1
Z π/3
(v) csc2 θ d θ
π/4
Z π/4 1 + cos2 θ
(w) dθ
0 cos2 θ
Z π/3 sin θ + sin θ tan2 θ
(x) dθ
0 sec2 θ

Dr. Angel Barria Comicheo