Area Of The Region Bounded By The Curves  Concepts & Shortcuts

Statutory warning : It's always best to sketch the curves than mugging up the formulae.
Some useful formulae:
Area bounded by the curves ax +/ m  = p and by +/ n = q is 4pq/ab sq units
Area bounded by ax +/ m + by +/ n = k is 2k^2/(ab)
Area bounded by ax + by = k and ax  by = k is 2k^2/ab
Area bounded by ax + by + ax  by = k is k^2/(ab)We will see in detail how we got these fancy formulas and how easy it is to derive them whenever you need it!
Area of a square with diagonal d = d^2/2
Area of a rhombus with diagonal d1 and d2 = (d1 x d2)/2
Area of parallelogram = base x heightSome important graphs
x = k (say k = 2) will give 2 lines parallel to y axis. One of which represents x = 2 and the other x =  2
Similarly y = k will give 2 lines parallel to x axis. One of which represents y = 2 and the other y =  2So x = 2 AND y = 2 we will give us a square with side = 2k as shown below.
Now what about x + 3 = 2 AND y + 1 = 2 ?
x + 3 = 2 => x = 1 or x = 5 (4 units)
y + 1 = 2 => y = 1 or y = 3 ( 4 units)
So this should also give a square of side 4 unitsGraph is as below
One interesting thing here is even if we plot x + 6 = 2 and y + 8 = 2
x + 6 = 2 => x = 8 or x = 4 (4 units)
y + 8 = 2 => y = 6 or y = 10 (4 units)
It will still give a square of side 4 units.Or say if we plot x  3 = 2 and y + 4 = 2
x  3 = 2 => x = 1 or x = 5 (4 units)
y + 8 = 2 => y = 6 or y = 10 (4 units)
again, a square of side 4 units.So all the values of x +/ a = k and y +/ b = k will yield a square of side 2k units. (definitely in different coordinates depending on a and b)
What if the value of k is different. Like x + 3 = 1 and y  1 = 2 ?
x + 3 = 1 => x = 2 or x = 4 (2 units)
y  1 = 2 => y = 3 or y = 1 (4 units)
A rectangle with sides 2 and 4.
We get a rectangle instead of square. That's it. Graph will be likeWe can say in general ax +/ m  = p and by +/ n = q will plot a rectangle with side 2p/a and 2q/b.
Hence area = 4pq/ab.Now we will see another important type.
x + y = 2 will give a square with diagonal = 2k as shown below.
Here also if you plot any x +/ a + y +/ b = 2 it will still yield a square with diagonal = 2k.
So in general, x +/ a + y +/ b = k will yield a square of diagonal 2k.
What about 3x + 4y = 12 ?
Graph is as below
here it is a rhombus with diagonals 6 and 8 (which is nothing but 12 x 2/3 and 12 x 2/4, where 3 and 4 are our coefficients). Area here is (2 x 12^2)/(3x4)
Here also if you try something like 3x + 6 + 4y  4 = 12, it will plot the same shape (a rhombus with diagonals as 6 and 8 and the only change will be the position of the rhombus in the xy plane which won't alter the area)
So in general, Area of ax +/ m + by +/ n = k is 2k^2/ab
We saw the graph of x + y = k, what about the graph of x  y = k ?
For example graph of x  y = 3 is as below
means it won't bound any region in the xy plane.
what about x  y = k type ?
For example, graph of x  y = 3 looks like
so this one also won't bound any region in the xy plane (good, lesser formulas!)
x + y = k case is also same. for example x + y = 3 won't bound any region in the xy plane. Graph is as below
What if we combine both ? i.e x + y = 3 and x  y = 3. Can you guess from the above graphs how it would turn out ?
Bounded region is as below
A square with diagonal as 6.
For example, the graph of 3x + 2y = 6 and 3x  2y = 6 is shown as below
So in general, if we plot ax + by = k and ax  by = k, we will get a rhombus with diagonals 2k/a and 2k/b.
Okay. so now we can solve the graph for x + y = 4 and x  y = 4. But what about x + y + x  y = 4 ?
Graph is as below
We can see that the graph will give a square of side = k
so area = k^2 = 16what would be the graph of 2x + y + 2x  y = 8 ?
Graph will be like
Rectangle with side 8 and 4. Area = 8 x 4 = 32
If you see this, the sides are nothing but 8/2 and 8/1. Where 2 and 1 are nothing by coefficients of x and y
So we can say that the area covered by the graph ax + by + ax  by = k is k/a * k/b = k^2/ab
How to plot the graph of a line, say 3x + 2y = 6 ?
when x = 0, y = 3
when y = 0, x = 2Graph is as below
x^2 + y^2 = r^2 is the equation of a circle with radius = r and center at origin.
For example, x^2 + y^2 = 9 will plot a circle as below
We have compiled some questions from this topic for your practice and can be referred at Question Bank  Area Of The Region Bounded By Curves
Share the formulas/concepts which we missed out and point out errors (if any).