**Q:- Find the area between the curves : x=1-y^2 and x=y^2-1**

**Ans:- **Lets say the following code is in file named as define1.sage. Go to terminal and type sage. You will get sage prompt if sage is installed on PC. type:

attach define1.sage** **
var('y')
b=1-y^2
r=y^2-1
e=solve(b==r,y)
print "Points of intersection"
print e
d=integrate(b-r,-1,1)
print "Area under two curves is:"
print d
c=plot(b,(y,-1,1),color='blue')
c+=plot(r,(y,-1,1),color='red')
L = [[-1 + i/10, b(-1 + i/10)] for i in range(21)]
M = [[-1 + i/10, r(-1 + i/10)] for i in range(21)]
pfill = polygon(L+M, color='yellow')
show(c+pfill)

**Output :**

Points of intersection

[

x== -1,

x== 1

]

Area under two curves:

8/3

**Note:** If on running program image is not provoked then replace show(c+pfill) with two lines q=c+pfill and q.save(‘filename.png’). With this image gets saved in home folder with name filename.png.

**Q:- Find the area between curves: y=4x-x^2 and y=x**

var('x')
b=4*x-x^2
r=x
e=solve(b==r,x)
print e
d=integrate(b-r,0,3)
print d
c=plot(b,(x,0,3),color='blue')
c+=plot(r,(x,0,3),color='red')
show(c,aspect_ratio=1)

**Output:**

[

y == 0,

y == 3

]

9/2

**Q:- Find the area between curve and x-ais, where curve is :y=x^2.**

**Ans:-**

a = -1; b = 1
f = lambda x: x^2
d=integrate(f(x),a,b)
print "Area under curve and x-axis is:"
print d
Lb = [[b,f(b)], [b,0], [a,0], [a,f(a)]]
Lf = [[i/20, f(i/20)] for i in xrange(20*a, 20*b+1)]
P = polygon(Lf+Lb, rgbcolor=(0.2,0.8,0))
Q = plot(f(x), x, a-0.5, b+0.5)
show(P+Q)

**Output:**

Area under curve and x-axis is:

2/3

**Q:- Find the area between two curves: y=0 and y=3*(x^3-x)**

**Ans:-**

var('x')
b=0
r=3*(x^3-x)
e=solve(b==r,x)
print "Points of intersection:"
print e
d=integrate(r-b,-1,0)+integrate(b-r,0,1)
print "Area between curves is:"
print d
c=plot(b,(x,-1,1),color='blue')
c+=plot(r,(x,-1,1),color='red')
show(c,aspect_ratio=1)

**Output:**

Points of intersection:

[

x == -1,

x == 1,

x == 0

]

Area between curves is:

3/2

**Q:- Find the area between curves y=x^2 and y=x^3**

**Ans:-**

var('x')
b=x^2
r=x^3
e=solve(b==r,x)
print "Points of intersection:"
print e
d=integrate(b-r,0,1)
print "Area between curves is:"
print d
text("x^2",(0.5,0.5))
c=plot(b,(x,-1,1),color='blue')
c+=plot(r,(x,-1,1),color='red')
t3 = text("y=x^2" ,(0.5,0.5), rgbcolor=(0,0,1))
show(c+t3,aspect_ratio=1)

**Output:**

Points of intersection:

[

x == 0,

x == 1

]

Area between curves is

1/12

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