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A past exam from math 155 at simon fraser university, held on 7 february 2007. It includes instructions for the exam and six math problems covering topics such as calculus, definite integrals, and volumes of solids of revolution.
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SIMON FRASER UNIVERSITY
MATH 155 Midterm 1
7 February 2007, 08:30–09:
Last Name
Given Name(s)
Student #
Signature
INSTRUCTIONS
Question Maximum Score 1 7
2 8
3 5
4 5
5 6
6 8
Total 39
Assume that all functions are continuous in the intervals of inte- gration.
A statement containing general constants and/or functions (f, g, a, b, c, n) is true if and only if it holds for all admissible choices of these constants and/or functions.
[1] (a)
∑n k=1(k^ + 1) =^
n(n + 3) 2
[1] (b)
∫ (^) b a f^ (x)g(x)^ dx^ =
(∫ (^) b a f^ (x)^ dx
(∫ (^) b a g(x)^ dx
[1] (c) If
∫ (^) b a f^ (x)^ dx^ ≥^0 , then^ f^ (x)^ ≥^0 for all^ x^ in^ [a, b].
[1] (d) If f (x) ≥ c for all x in [a, b], then
∫ (^) b a f^ (x)^ dx^ ≥^ c.
[1] (e)
0 x^ sin^ x dx^ ≤^
[1] (f)
∫ (^) b a [f^ (x)^ −^ g(x)]^ dx^ =
(∫ (^) b a f^ (x)^ dx
(∫ (^) b a g(x)^ dx
[1] (g)
∫ (^) b a f^ (x)^ dx^ =^
(∫ (^) c a f^ (x)^ dx
(∫ (^) c b f^ (x)^ dx
[5] 3. Determine the average value of f (x) =
1 − x^2
in the interval [0, 1 /2].
[5] 4. Express the area between the curves y = 2x^2 + 4 and y = 7 − x^2 as a definite integral. DO NOT EVALUATE THE INTEGRAL.
[3] 5. (a) Describe the right-circular cone with base radius r and height h as a solid of revolution.
[3] (b) Use the formula for the volume of a solid of revolution to express the volume of the right-circular cone with base radius r and height h as a definite integral. DO NOT EVALUATE THE INTEGRAL.