MATH 1A, 8-9am section
Quiz 10, April 20
Name:
Student ID:
You have 20 min to complete this quiz. Please state your answers clearly and precisely.
1. Given that 03 f (x)dx = 7 and 13 f (x)dx = 3, find 01 f (x)dx (3
points)
R
R
R
Solution: We know that 03 f (x)dx = 01 f (x)dx + 13 f (x)dx, so
R1
R3
R3
0 f (x)dx = 0 f (x)dx − 1 f (x)dx = 7 − 3 = 4
R
R
R
R √
2. Evaluate the integral 02 4 − x2 dx by interpreting it as an area (3
points)
√
Solution: The function f (x) = 4 − x2 for 0 ≤ x ≤ 2 defines a
quarter of the circle with radius 2 and center at the origin. So if
we interprete
the intergral as the area under the curve, the value
R2 √
will be 0 4 − x2 dx = 14 π22 = π
3. Using the definition of the integral, evalute 01 (1 + x2 )dx. You may
P
want to use that ni=1 i2 = n(n+1)(2n+1)
(4 points)
6
2 R1
P
1
2
n
Solution: 0 (1 + x )dx = limn→∞ i=1 1 + ni
n =
R
h P
i
1
1 Pn
n
2
1
+
(
)
i
i=1
i=1
n3
i
h n
limn→∞ 1 + 61 (1 + n1 )(2 + n1 ) = 1
limn→∞
= limn→∞ 1 +
+
1
3
1 n(n+1)(2n+1)
n3
6
=