Section 5.7 #71

Maple Graphs and Calculations

>	with(plots):

Warning, the name changecoords has been redefined

>	plot(arcsin(x),x=0..1,filled=true);

>	Int(arcsin(x),x=0..1);

Maple is smart enough to be able to compute this exactly.

>	int(arcsin(x),x=0..1);

Note that y = arcsin(x) means that x = sin(y).

>	implicitplot(x=sin(y),x=0..1,y=0..Pi/2,thickness=3);

We can observe that the area of the region pictured above would be the area of a rectangle whose dimensions are  by 1 minus the area of the region pictured below.

>	plot(sin(x),x=0..Pi/2,filled=true);

In the picture constructed below I graphed the rectangular area in blue and graphed the sine area over it in red (the default color).

>	RectangleArea:=plot(1,x=0..Pi/2,color=blue,filled=true):

>	SineArea:=plot(sin(x),x=0..Pi/2,filled=true):

>	display(SineArea,RectangleArea);

Thus we get the result below which would involve "easy" integration.

>	Int(arcsin(x),x=0..1) = Pi/2 - Int(sin(x),x=0..Pi/2);

>	Pi/2-int(sin(x),x=0..Pi/2);

>