Application of Integrals

Area Under Simple Curves

Area Under Curves: Have you ever shared your umbrella with a companion? Supposing that the two of you are very accommodating, then the extent to which you will get drenched in the rain will obviously depend on the size of the umbrella and your companion! Now if you want to be precise about it, in mathematical terms you can simply discuss your situation with reference to the area under the umbrella available to the two of you.

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If the area under the umbrella is less than the area you and your partner need to stand in, both of you will get drenched from the sides, whereas if the area under the umbrella is greater than the combined area of you and your partner, both of you would be shielded from the rain! These types of problems fall under the category of the analysis of area under curves, which is going to be our topic of discussion right now.

We will talk about the area under curves in one dimension in this section. You could also talk about areas under curves of higher dimensions but you’ll encounter such problems only when you do some higher studies, for example –  the umbrella we just talked about! (which is a 3-dimensional figure).

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Area Under Curves

Before beginning the discussion, it should be quite clear that it makes sense to talk about the area under a curve only when you have a graph of that curve. Now let us have a function of x given as y = f(x). Shown below is such a general curve whose values can clearly be positive as well as negative, depending on the value of x.
Area under curves

It is evident from the graph that the area below the curve is actually unbounded! In other terms, if we don’t specify the values of x within which the area under the curve has to be calculated; then our answer is indeterminable. In the fig. below, you can see how the boundaries of the area available under the curve are set by the limiting values of x.
Area under curves

                                                                   Learn Area Between Two Curves here in detail. 

Calculation Method

Step 1: Form a rectangular strip of height/length = f(x0) and breadth = dx as shown in the figure below.

  • You can consider the rectangle to be centered at the value x = x0
  • dx is an infinitesimally small value which could be taken equal to the difference in the x-coordinates on which the sides of the rectangle are placed.

Area under curves

Step 2: Move the strip under the curve, beginning from the lower bound of x i.e. x = a; and terminating at the upper bound of x i.e. x = b, changing the value of x0 at each point but retaining the same value of dx throughout. Place all of these strips adjacent to each other and get a resultant figure such as:

Area under curves

Step 3: The area dA of a single rectangular strip = length × breadth
$$ {dA = f(x_0) \times dx} $$
This is known as the Differential/Elementary Area.

Step 4: The total area A under the curve can be approximately obtained by summing over the areas of all the rectangular strips.
$$ {A = \sum\limits_{x_0 = a}^{x_0 = b} dA} $$
Using the value of dA from a previous step:
$$ {A = \sum\limits_{x_0 = a}^{x_0 = b} f(x_0)dx} $$

Step 5: If dx → 0, the summation can be converted to an integral. Then we have;
$$ {A = \int_{a}^{b} f(x)dx} $$

Here we have used a Reimann Sum to calculate this area under the curve y = f(x) i.e. approximated an integral by using a finite sum; since the number of the rectangular strips was finite but taking that number → ∞ (the same as taking dx → 0), we converted the sum to an integral. Now we are ready to discuss the formal definition for the area under curves.

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Area Under Simple Curves

Definition of Area Under Curves

The area A under the curve f(x) bounded by x = a and x = b is given by: $$ {A = \int_{a}^{b} f(x)dx} $$

  • If the area between two bounding values of x on the graph, lies above the x-axis; its sign is taken to be positive.
  • If the area between two bounding values of x on the graph, lies below the x-axis; its sign is taken to be negative.

Area under curves

This should make it clear to you that the name: ‘Area under the curve’ is not meant to be taken literally. It is just a mathematical definition. Now let us work out a problem based on the formula that we have just derived.

An Alternate Analysis

Sometimes it is easier to express/plot a given function y = f(x) as x = f(y). It might also be the case when the function in the form of x = f(y) is more easily integrable as compared to y = f(x). In such cases, the area under a curve would be the one with respect to the y-axis. The figure given below would make things clear to you.
Area under curves

You can see that here by constructing horizontal rectangular strips of length f(y0) and breadth dy, one can derive another form of the formula for the area under a curve. $$ {A = \int_{y = b}^{y = a} f(y)dy} $$

In similarity with our original case:

  • If the area between two bounding values of y on the graph, lies to the right side of the y-axis; its sign is taken to be positive.
  • If the area between two bounding values of y on the graph, lies to the left side of the y-axis; its sign is taken to be negative.

Note: Sometimes one is asked to find the total area bounded by a given curve. In that case, the definite integral could give you the result which is less than what is expected. For example- try calculating the area under the curve y = sin x from x = 0 to x = Π/2. You’ll get the result = 0.

Does the total area under the curve look 0 to you? Nope! Thus, the correct evaluation, in that case, is to take a modulus of the negative values of the area obtained under the curve i.e. convert all the negative areas to positive and add!

                                                 Also, let us study Area Under the Curve Bounder by a Line in detail. 

Solved Examples for You

Question: Calculate the area under the curve \({ y = \frac{1}{x^2}} \) in the domain x = 1 to x = 2.

Solution- The domain of x lies n the first quadrant only. Thus we need to be concerned with the graph of the given function in the first quadrant only. As discussed in the steps above, we can visualize the applicability of our formula by forming vertical rectangular strips under the curve, and add over all of them to get the area. Shown below is the graph for this problem:

Area under curves

The area under the curve in the region shown above can be given by:
$$ {A = \int_{x = 1}^{x = 2} \frac{1}{x^2} dx} $$
Solving the integral, we can get
$${A = [- \frac{1}{x}]_{x = 1}^{x = 2}} $$
$${A = – \frac{1}{2} – ( – 1)} { = \frac{1}{2}} $$

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