Calculus I Name:
TA/classroom: Student ID:
Quiz 10
Dec. 26, 2007
1. (5 pts) Use Riemann Sums to compute the given definite integral
∫ 1
0
x2dx
Set f (x) = x2, ∆x = 1N−0 and xi = 0 + i∆x = Ni. Then
AN =
∑N i=1
f (xi)∆x =
∑N i=1
( i N)2 1
N = 1 N3
∑N i=1
i2 = 1 N3
N (N + 1)(2N + 1) 6
∫ 1
0
x2dx = lim
N→∞
1 N3
N (N + 1)(2N + 1)
6 = 1
3
2. (5 pts) Use the Fundamental Theorem to compute the given definite integral
∫ 1 0
x2dx = 1
3x3|10 = 1 3
3. (10 pts) Given F (x) = ∫x2 x
√t2+ 1 dt, use the Fundamental Theorem to compute F0(x)
F0(x) =√
x4+ 1(x2)0 −√
x2+ 1(x)0 = 2x√
x4+ 10−√
x2+ 1.
or, let G(x) =∫x 0
√t2+ 1 dt, then we have F (x) = G(x2)−G(x) and, by Fundamental Theorem of Calculus, G0(x) =√
x2 + 1.
Thus
F0(x) = (G(x2)− G(x))0 = G0(x2)(x2)0− G0(x) = 2x√
x4+ 1−√
x2+ 1.
. .
• Theorem 1.1 If n is any positive integer and c is any constant, then
∑n i=1
c = cn,
∑n i=1
i = n(n + 1)
2 ,
∑n i=1
i2 = n(n + 1)(2n + 1) 6
• Fundamental Theorem of Calculus Part I: If f is continuous on [a, b] and F (x) is any antiderivative of f , then∫b
a f (x)dx = F (b)− F (a).
• Fundamental Theorem of Calculus, Part II: If f is continuous on [a, b] and F (x) =∫ x
a
f (t)dt, then F0(x) = f (x), on [a, b].