Area Between Curves

September 20, 1999
Section 6.1

Problem 37. Find the area of the region bounded by and with in the interval .

> plot({2*sin(x),tan(x)},x=-Pi/3..Pi/3);

> Area37 := 2*int(2*sin(x)-tan(x),x=0..Pi/3);

> evalf(Area37);

Problem 41. Graph the region bounded by the graphs of the functions , , and in the interval .

> plot(2*sin(x)+sin(2*x),x=0..Pi);

> Area41 := int(2*sin(x)+sin(2*x),x=0..Pi);

Problem 63 . Graph the region inside the square of width 4 centered at the origin, where the points in the region are closer to the
center of the box than to the side of the box. Then determine the area of the region.

> eq1 := sqrt(x^2 + y^2) = 2 - y; eq1a := solve(eq1,y);

> eq2 := sqrt(x^2 + y^2) = 2 - x; eq2a := solve(eq2,x);

> solve(eq1a = x,x);

> plot({[t,2,t=-2..2],[2,t,t=-2..2],[t,-2,t=-2..2],[-2,t,t=-2..2], [t,1-t^2/4,t=-2*(sqrt(2)-1)..2*(sqrt(2)-1)],
[1-t^2/4,t,t=-2*(sqrt(2)-1)..2*(sqrt(2)-1)],
[t,-1+t^2/4,t=-2*(sqrt(2)-1)..2*(sqrt(2)-1)],
[-1+t^2/4,t,t=-2*(sqrt(2)-1)..2*(sqrt(2)-1)]},
color=[red, red, red, red, black, black, black, black], axes=NORMAL, scaling=CONSTRAINED);

> Area63 := 8*int(1-x^2/4-x,x=0..2*(sqrt(2)-1));

> Area63 := simplify(Area63);

> evalf(Area63);

>