CHAPTER 7 - Techniques of Integration

Section 7.1, page560

Problem 32

>	Int(1/(r*sqrt(r^2-9)),r)=int(1/(r*sqrt(r^2-9)),r);

Problem 40

>	Int(1/sqrt(2*theta-theta^2),theta)=int(1/sqrt(2*theta-theta^2),theta);

Problem 66

>	Int(sqrt(1+cos(t)),t=-Pi..0)=int(sqrt(1+cos(t)),t=-Pi..0);

Section 7.2, page 567

Problem 14

>	Int((r^2+r+1)*exp(r),r)=int((r^2+r+1)*exp(r),r);

Problem 41

(a) Average value:

>	avg:=simplify(1/(2*Pi-0)*int(2*exp(-t)*cos(t),t=0..2*Pi));

>	plot({avg,2*exp(-t)*cos(t)},t=0..2*Pi,color=blue,thickness=2);

Section 7.3, page 576

Problem 11

We could just do the integral, but we want to point out that Maple can show you a partial fractions decomposition as follows:

>	convert((x+4)/(x^2+5*x-6),parfrac,x);

>	Int((x+4)/(x^2+5*x-6),x)=int((x+4)/(x^2+5*x-6),x);

Problem 48

We'll find the centroid of the region. First, the area:

>	f:=(4*x^2+13*x-9)/(x^3+2*x^2-3*x);

>	area:=int(f,x=3..5);

Now for the moments around the y  and x axis:

>	My:=int(x*f,x=3..5);

>	Mx:=int(f/2*f,x=3..5);

>	centroid:=[My/area,Mx/area];

>	evalf(centroid);

Section 7.4, page 582

Problem 24

>	Int(sqrt(1-x^2)/x^4,x)=int(sqrt(1-x^2)/x^4,x);

Section 7.6, page 603

Problem 21

>	Int(theta*exp(theta),theta=-infinty..0)=int(theta*exp(theta),theta=-infinity..0);

Problem 71

First graph the region that we are going to revolve around the y-axis:

>	plot(exp(-x),x=0..10,color=blue,thickness=2);

We'll calculate the volume by shells -- for each x from 0 to infinity, the shell has radius x, thickness dx and height .

>	volume:=int(2*Pi*x*exp(-x),x=0..infinity);