Quiz 2018, answers PDF

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Math 101 - 202 Quiz #1 (January 16, 2012) Show all your work. Use back of page if necessary. Calculators are not allowed. Last Name:

First Name:

Student No.:

R1

(x3 +1) dx by taking the limit of the Riemann sum obtained by using n equal-length n X i3 = n2 (n + 1)2 /4. No subintervals and the sampling point x∗i = xi (i.e. Rn ). You may use the formula

1: (5 marks) Evaluate

0

i=1

credit for a solution that uses the antiderivative.

 3  Pn Pn 3 (x3 + 1) dx = limn→∞ i=1 1n ni3 + 1 = limn→∞ n−4 i=1 i + 1 = 54 . R2 √ 2: (3 marks) Evaluate 0 4 − x2 dx by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high-school geometry. Solution:

R1 0

Solution: Graph the function; it is 1/4 of a circle of radius 2. The area is π .     n X π iπ iπ 3: (2 marks) Express lim sin cos as a definite integral. Do not evaluate the integral. n→∞ n n n i=1

Solution:

Rπ 0

sin(x) cos(x) dx....