Download (1) Evaluate the following expressions. (a) If x = 2 tan

MATH 231 WORKSHEET 4
(1) Evaluate the following expressions.
(a) If x = 2 tan θ, evaluate sin θ.
(b) If x = 2 sin θ + 1, evaluate sec θ.
(2) Evaluate√the following integrals.
Z
4x2 − 1
dx
(a)
x
Z
4
(b)
dx
2
2
Z (x + 16)
1
(c)
dx (Hint: ALGEBRA!)
2
Z 3 − 2x − x
1
(d)
3 dx
(4 − x2 ) 2
x2
y2
(3) (a) Compute the area of the ellipse 2 + 2 = 1. (Hint: it may help to draw the ellipse.)
a
b
(b) What happens when a = b = r?
(4) A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length
L. The man keeps the rope straight and taut, and the path followed by the boat is a curve called a
tractrix. The tractrix has the property that the rope is always tangent to the curve, as shown in the
figure.
(a) If the path the boat follows is given by the function y(x), compute
(b) Determine the function y = f (x).
Date: January 28, 2010.
1
dy
dx .
Table of Trigonometric Substitutions
Expression
a2 − x2
a2 + x2
x2 − a2
Substitution
x = a sin θ
x = a tan θ
x = a sec θ
2
Identity
1 − sin2 θ = cos2 θ
1 + tan2 θ = sec2 θ
sec2 θ − 1 = tan2 θ