Download MAS110 Problems for Chapter 7: Integration

MAS110 Problems for Chapter 7: Integration
1. By slicing the area into N thin pieces approximated by rectangles, and proceeding as for
RX
in the first integration lecture, find an alternative proof that 0 x dx = X 2 /2.
RX
0
x2 dx
RX
3
2. Strictly speaking, in the lecture we found an overestimate: 0 x2 dx < X6 1 + N1 2 + N1 .
Some hand-waving was required to believe that as N → ∞ the limit of the overestimate is the
true area. Here is a more careful approach.
rectangles that cut underneath the curve
By Rusing
X 2
1
1
X3
obtain an underestimate: 6 1 − N 2 − N < 0 x dx. The true area is definitely sandwiched
between the underestimate and the overestimate. Now what happens as N → ∞?
3. Here is another example of using Fermat’s method of subdividing an interval to calculate integrals.
Let X > 0 be a real number.
(a) Sketch a picture to illustrate an overestimate the area of the region between the x-axis and
the curve y = x1/2 over the interval [0, X] by using the points . . . , a3 X, a2 X, aX, X, for
some 0 < a < 1. Calculate the overestimate.
RX
(b) Letting a → 1 calculate 0 x1/2 dx.
RX
(c) If you want to you can work out 0 xm/n dx for integers m, n 6= 0 with m/n > −1 using
the same method.
4. Let n be a positive integer and X > 0 a real number. Calculate
your answer geometrically in terms of two areas.
RX
0
xn dx +
R Xn
0
x1/n dx. Explain
5. Assume all functions that appear Zare sufficiently well-behaved. What does the Fundamental The
d x
orem of Calculus say about
f (t) dt ? Combining with the chain rule, show that
dx 0
d
dx
d
Obtain a similar relation for
dx
Z
Z
h(x)
f (t) dt = f (h(x))h0 (x).
0
h(x)
f (t) dt .
g(x)
Z
Z
6. Give an example to illustrate the fact that in general
Z
7. Calculate
2π
Z
x sin x dx and
0
u(x)v(x) dx 6=
Z
u(x) dx
v(x) dx .
2π
x2 cos 2x dx.
0
8. We knowZhow to integrate cos2 x using the formula cos2 x = (1 + cos 2x)/2. As an alternative,
π
evaluate
cos2 x dx by writing cos2 x = cos x cos x, and integrating by parts.
0
1
9. Let f and g be functions selected from the set
{1, cos x, sin x, cos 2x, sin 2x, cos 3x, sin 3x, . . .}.
Integrating by parts, and using the formulas for sin2 θ


Z 2π
0
f (x)g(x) dx = 2π

0

π
1
Z
x2
10. Differentiate e . Evaluate the integrals
and cos2 θ, show that
if f 6= g;
if f = g = 1;
if f = g 6= 1.
Z
x2
0
Z
1
11. Evaluate the integrals
0
x2
dx,
1 + x3
Z
12. Integrating by parts, evaluate
Z
Z
0
1
1
2
x3 ex dx.
xe dx and
0
x2
dx and
1 + x2
Z
1
0
x3
dx.
1 + x2
tan−1 x dx.
π
ex cos x dx.
13. Evaluate
0
d
(sec x) = sec x tan x. By
14. Show that if −π/2 < x < π/2 then sec x + tan x > 0. ZShow that dx
making the substitution u = sec x + tan x, show that
sec x dx = ln(sec x + tan x) + C (if
−π/2 < x < π/2).
d
dx
2
3
2
(tan x) = sec
Z x? By writing sec x = sec x sec x and integrating by parts, evaluate
the indefinite integral sec3 x dx.
15. Why is
Z
16. By making a trigonometric substitution, show that
0
Z
2
17. Evaluate
x2
1
Z
x
√
√
1
dt = ln(x + x2 + 1).
1 + t2
1
dx.
+ 2x
1
1
√
dx. At some point you might like to use a trigonometric substitution
x2 + 2x
0
involving sec θ.
18. Evaluate
19. Evaluate the following improper integrals.
Z ∞
Z ∞
ln(x)
3 −x4
dx;
(b)
(a)
xe
dx;
x2
0
1
2
Z
(c)
0
∞
xn e−x dx where n ∈ N.