sheet23.pdf

course: CalculusII

March5,2007

Problemsheet:Riemannintegral

1.Evaluatetheintegrals

(a)

0 sinxdx,(b)

2 R

|1−x|dx,(c)

2 R

1+"cosx (0"<1)(d)

e R

|lnx|dx

1/2

2.ConstructingRiemannsumsofappropriatelychosenfunctionsﬁndthefollowinglimits:

(a)lim

n!1

n 2 + 2 n 2 +...+ n−1

, (b)lim

n!1

n+1 + 1

n+2 +...+ 1

n 2

n+n

,(d)lim

n!1

(c)lim

n!1

n 2 +1 + n

n 2 +2 +...+ n

sin n +sin 2 n +...+sin (n−1)

n 2 +n 2

1+ n

(e)lim

n!1

1 p +2 p +...+n p

n p +1 (p>0) (f)lim

1+ 1 n +

1+ 2 n +...+

n!1

3.Findtheintegralsusingmethodsofsubstitutionsandintegratingbyparts:

(a)

ln2

xe −x dx(b)

1 R

arccosxdx(c)

a R

x 2 p a 2 −x 2 dx(d)

ln2

p e x −1dx

4.Evaluatetheintegrals

(

( x 0xt

x 2 0x1

2−x1<x2 (b)

2 R

1 R

(a)

f(x)dxf(x)=

t 1−x

1−t

tx1

5.FindintegralsandplotagraphofthefunctionI=I()

sin 2 x

1+2cosx+ 2 dx

6.Showthatiffisacontinuousfunctionthen

1 R

x|x−|dx(b)I=

(a)

b R

a f(x)dx=(b−a)

1 R

0 f(a+(b−a)x)dx(b)

0 x 3 f(x 2 )dx= 1

a 2 R

0 xf(x)dx(a>0)

7.Assumingthatfiscontinuouson[0,1]showthat

(a)

f(sinx)dx=

f(cosx)dx(b)

xf(sinx)dx=

f(sinx)dx

8.FindtheexplicitformofthefunctionF(x)

(a)F(x)=

x R

(|t−1|+|t+1|)dt(x0) (b)F(x)=

x R

||t−1|−2|dt

9.Solvetheeuqations:(a)

x R

p 2

t p t 2 −1 =

12 (b)

x R

p e t −1 =

ln2

10.Derivearecursiverelationfortheintegrals:

(a)I n =

sin n xdx(b)I n =

1 R

(1−x 2 ) n dx(c)I n =

1 R

p 1−x 2 dx(d)I n =

x n

1 R

x m (lnx) n dx

11.Evaluatethefollowingintegrals:

(a)

3 R

sgn(x−x 3 )dx(b)

2 R

[e x ]dx(c)

xsgn(cosx)dx(d)

1 R

sgn(sin(lnx))dx

(a)I=

a R

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