Application of Integrals

Area Bounded by a Curve and a Line

Area Under the Curve Bounder by a Line: The method of calculation of the area under simple curves laid down the foundation for solving various complex problems using the same logic. A class of such problems is the calculation of the area under the curve bounded by a line. It is a very straightforward topic to understand, so we will jump straight into it!

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Formulation of Area Under the Curve Bounded by a Line

Given a curve C: y = f(x) and a straight line T: y = mx + c. The first step is to plot the area under the curve and the straight line on the same graph. Various cases may be possible:

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Bounded Regions

  • The straight line intersects the curve at two points and forms a bounded region.

area bounded by a curve and a line

  • The straight line intersects the curve at three points and forms more than 1 bounded region. The total area has a finite value.

area bounded by a curve and a line

  • The straight line intersects the curve at an infinite number of points and forms an infinite number of bounded regions. The total area is not finite here.

area bounded by a curve and a line

Regions with No Bounds

  • The straight line touches the curve at 1 point. Similarly, the line may touch the curve at multiple points. In such cases, no bounded region exists.

area bounded by a curve and a line

  • The straight line neither intersects nor touches the curve at any point.

area bounded by a curve and a line

Unbounded Regions

  • The straight line and the curve form an unbounded region.

area under the curve

Now we will discuss how to calculate the area bounded by the curve and the straight line.

Calculation of Area Under the Curve Bounded by a Line

Let the graph of the curve and the straight line look something like this:

area under the curve

Clearly, we need to calculate the area of the mentioned region in the graph. This region can be viewed as the region common to the ‘Area under the curve’ for both, the straight line as well as the given curve. This simplifies our calculation tremendously. For example, here the straight line T has more area under it than the curve C.

So, the area common to both of them can be found out by subtracting the area under the straight line T from the area under the curve C. The bounding values of x for the calculation of the area under the curves can be found by solving the simultaneous equations for the coordinates of the points of intersection between the straight line and the curve.

Let those points have x-coordinates x1 and x2. The area under the curves (say y = f(x)) can be calculated using the formula:

\( {A = \int_{x_1}^{x_2} f(x)dx} \)

Let the area under the given curve C turn out to be AC and let the area under the given straight line T turn out to be AT. Then the area common to both the curves, in this case, will be A = AT – AC. A general formula for writing down the area for this case could be given as: $$ {A = \int_{x_1}^{x_2} [f(x) – g(x)]dx} $$
where f(x): equation of the straight line and g(x): equation of the curve. Similarly, other cases of the intersection of a straight line and a curve can be analyzed and a general formula could be developed for them.

Conventions

For the formula derived to be applicable, the following conventions will have to be followed.

  • If the straight line is above the curve in the bounded region, the equation of the straight line is f(x) and the equation of the curve is g(x) and vice-versa.
  • The area above the y-axis is taken with a positive sign.
  • The area below the y-axis is taken with a negative sign.

If the curve can easily be expressed in the form of x = f(y); then we must express the straight line also in the form of x = g(y). In such a case, we could employ the formula:$$ {A = \int_{y_1}^{y_2} [f(y) – g(y)]dy} $$

The conventions to be followed in this case would be:

  • If the straight line is to the right of the curve in the bounded region, the equation of the straight line is f(x) and the equation of the curve is g(x) and vice-versa.
  • The area on the left side of the x-axis is taken with a positive sign.
  • The area on the right side of the x-axis is taken with a negative sign.

This brings us to the end of our discussion. Let us now look at a solved problem.

Solved Examples for You

Question: Find the area bounded by the curve y = x2 + 2 and straight line y = x + 3.

Solution: The first step is the calculation of the coordinates of the intersection points M and N. We must solve the equations y = x2 + 2 and y = x + 3 simultaneously for it. Put the value of y in the equation of the curve to get:

x + 3 = x2 + 2
→  x2 – x + 1 = 0

This is a quadratic equation which has two real roots. Solving for them, you can get x1 = -0.618 and x2 = 1.618. Put these values in the equation of the straight line to get the corresponding y-coordinates as y1 = 2.382 and y2 = 4.618. Let us look at it on the graph:

area under the curve

Now we have our bounded region, with the bounding values of x known to us. For the formula to be applicable:

  • Here f(x) = x + 3 and g(x) = x2 + 2.
  • Since the region is above the y-axis, its sign will be taken as positive.

Then the area A will be given as: $$ {A = \int_{-0.618}^{1.618} [(x + 3) – (x^2 + 2)] dx} $$
$$ {A = \int_{-0.618}^{1.618} [(- x^2 + x + 1)] dx} $$
$$ {A = [(- \frac{x^3}{3} +\frac{x^2}{2} + x)]_{-0.618}^{1.618}} $$
$${A = 1.8633} $$

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