Calculus - Standalone book
3rd Edition
ISBN: 9781464125263
Author: Jon Rogawski, Colin Adams
Publisher: W.H. Freeman & Co
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Chapter 6.1, Problem 24E
To determine
The area under two curves.
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3. (i) Let A, B, C, D, E be positive constants. Find the limit
n³ D
lim A(BC) +1 en 4 + E.
n→∞
[6 Marks]
(ii) Let X be a random variable with the standard normal distribution,
i.e. X has the probability density function
Let
1
f(x)
n3x2
Xn = 100(50 +X6) en³4, n = 1, 2,...,
3+1
(a) Find the limit lim∞o Xn, verify your answer.
Y = 100(50+X) 2x².
[6 Marks]
(b) Is Xn Y? Verify your answer.
[6 Marks]
(c) Is EY∞o? Verify your answer.
[6 Marks]
(d) Does lim∞ EXn
=
Elimno Xn? Why?
[6 Marks]
(e) Find lim∞ EXn.
[3 Marks]
3. (i) Using the definition of the line integral of a vector field, calculate the
line integral
L³
F.dy
of the vector field F: R² → R² given by
F(x, y) = (y, x),
and where the curve & is the unit semi-circle centred at the origin, located in
the upper half-plane and oriented in the anticlockwise direction.
Hint. Represent the curve y as the join of two curves y = 71 + 1/2 (see Example 8.9
in the Notes).
[20 Marks]
(ii) Calculate the same integral using Green's Theorem.
[10 Marks]
1. Evaluate the integral
↓ f(x, y)dxdy,
of function f R² →R over the domain DC R2, where:
f(x, y) = 2x + y
and D is the is the triangle with vertices (0, -1), (1,0) and (0,2).
Hint. Represent D in the form D = {(x, y) = R² : x = (a, b), g(x) < y < h(x)} for
some a
Chapter 6 Solutions
Calculus - Standalone book
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