Math 1210
Quiz 8
March 20th, 2014
This is a take-home quiz. It contains 2 ordinary questions (35 points) and
1 extra credit question (15 points). The value of every question is
indicated at the beginning of it. Please print this sheet and write your
answers in the space provided.
Name:
UID:
1. (15 points) Compute the following indefinite integrals.
R 2 2
dx
(i) (5 points) (x √+1)
x
Hint: Expand the numerator first.
(ii) (5 points)
R
√ 3y
2y 2 +5
dy
R
(iii) (5 points) sin x(1 + cos x)4 dx
Hint: Use the generalized power rule.
2. (20 points) Find the area under the curve y = x3 + x between x = 0 and x = 1 by
following the steps below:
Hint: The answer is 43 .
(i) (2 points) Subdivide the interval [0, 1] into n equal subintervals [x0 , x1 ], [x1 , x2 ], . . . , [xn−1 , xn ].
What is the length ∆x of every subinterval? For every k, write an expression for
xk .
(ii) (5 points) Sketch the rectangles that you will be using in the graph below and write
down an expression for the area of the rectangle over [xk , xk+1 ].
Figure 1: Question 2
(iii) (5 points) Find the sum A(Rn ) of the areas of the n rectangles.
2
Pn
n(n+1)
3
Hint: Remember that k=1 k =
.
2
(iv) (4 points) The area under the curve will be the limit A = limn→∞ A(Rn ). Find this
limit.
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R
(v) (4 points) Compute the indefinite integral F (x) = x3 + x dx (so that F 0 (x) =
x3 + x) and find F (1) − F (0). You should be obtaining the same result as in (iv).
You will understand why in section 4.4.
3. EXTRA CREDIT (15 points)
(i) (10 points) Consider the differential equation
dy
= −y 2 x(x2 + 2)4
dx
Use separation of variables in order to find the general solution of this equation.
(ii) (5 points) Consider the differential equation
dy
x + 3x2
=
dx
y2
Show that y(x) =
q
3
3 2
x
2
+ 3x3 + 216 is a solution of this equation.
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