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MATH1011 Quizzes

Quiz 11: One variable integration
Question 1 Questions
Evaluate 01(6x2 4x + 3)dx.
a)
4
 b)
-3
c)
3
 d)
5

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
01(6x2 4x + 3)dx = 2x3 2x2 + 3x 01 = 2 2 + 3 = 3.
Choice (d) is incorrect
Evaluate 0π 2 (sinx + cosx)dx.
a)
2
 b)
0
c)
π 2
 d)
-2

Choice (a) is correct!
0π 2 (sinx + cosx)dx = cosx + sinx0π 2 = cos π 2 + sin π 2 (cos0 + sin0) = 1 + 1 = 2.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate 0π 4 sin2xdx.
a)
1
 b)
1 2
c)
1 2
 d)
-1

Choice (a) is incorrect
Choice (b) is correct!
0π 4 sin2xdx = 1 2cos2x0π 4 = 1 2cos π 2 + 1 2cos0 = 1 2.
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate 03x 1 + x2 3 2 dx.
a)
32 5
 b)
31 5
c)
3 2
 d)
9 2

Choice (a) is incorrect
Choice (b) is correct!
03x 1 + x2 3 2 dx = 1 5(1 + x2)5 2 03 = 1 5 45 2 1 5 = 32 5 1 5 = 31 5 .
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate 023x2ex3 dx.
a)
e8 1
 b)
8e8
c)
e8
 d)
12e8 12

Choice (a) is correct!
023x2ex3 dx = ex3 02 = e8 e0 = e8 1.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Let A be the area under the curve f(x) = 16 x2 on the interval [0,4]. Dividing the interval into 4 subintervals and defining xi to be the midpoint of the ith interval (the area of the rectangle is thus f(xi) times the length of the interval), the best estimate for A is
a)
A = i=14(16 x i2) × 1 2, where xi = i 1 2
b)
A = i=14(16 x i2) × 1 2, where xi = i 1 2
c)
A = i=14(16 x i2), where xi = 2i 1 2
d)
A = i=03(16 x i2), where xi = 2i 1 2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The intervals are of length 1, and the area is calculated at 1 2, 3 2, 5 2 and 7 2.
Choice (d) is incorrect
Which of the following sums is the best estimate for A, the area under the curve f(x) = x3 + 2 on the interval [-1,2], divided into 6 subintervals and choosing xi as the right-endpoint of the ith interval ?
a)
A = i=06(x i3 + 2), where xi = i 3 2
b)
A = i=06(x i3 + 2) × 1 2, where xi = 2i 1 2
c)
A = i=16(x i3 + 2), where xi = 2i 3 2
d)
A = i=16(x i3 + 2) × 1 2, where xi = i 2 2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The intervals are of length 1 2, and the area is calculated at 1 2, 0, 1 2, 1, 3 2, 2
Find the indefinite integral 2t2(1 + t3)4dt.
a)
1 5(1 + t3)5 + C
 b)
2 5(1 + t3)5 + C
c)
2 3t3(1 + t3)5 + C
 d)
2 15(1 + t3)5 + C

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Let u = 1 + t3, du = 3t2dt. 2t2(1 + t3)4dt = 2 3u4du = 2 15u5 + C = 2 15(1 + t3)5 + C.
Find the indefinite integral sin3xcosxdx.
a)
sin4x + C
 b)
1 4sin4x + C
c)
3sin2x + C
 d)
1 4sin4x + C

Choice (a) is incorrect
Choice (b) is correct!
Let u = sinx, du = cosxdx. sin3xcosxdx = u3du = 1 4u4 + C = 1 4sin4x + C.
Choice (c) is incorrect
Choice (d) is incorrect
Find the indefinite integral
2t et + 4 (t2 + 4t + et + 1)2dt.
a)
1 t2 + 4t + et + 1 + C
 b)
1 3(t2 + 4t + et + 1)3 + C
c)
1 t2 + 4t + et + 1 + C
 d)
1 3(t2 + 4t + et + 1)3 + C

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Let u = t2 + 4t + et + 1, du = (2t + 4 et)dt. 2t et + 4 (t2 + 4t + et + 1)2dt = 1 u2du = u1 + C = 1 t2 + 4t + et + 1 + C.
Choice (d) is incorrect