Area Under Simple Curves

The first chapter of Application of Integration discusses how the integration methods you acquired in the previous chapters may be used in various phases. In this chapter, you’ll focus on locating areas under simple curves. On the Cartesian coordinate axis plane, these curves are drawn. As a result, the bounds will represent the two axes X and Y. The challenges you try to solve will reveal the boundaries of these axes. Continue reading to learn about the various techniques, which are divided into sections for simple comprehension.

The area under a curve between two points is found out by doing a definite integral between the two points. 

Let’s find the area swooped by the curve y = f(x), x-axis where ordinates are x = a and x = b.

Areas under simple curves are generally thought to be made up of a substantial amount of very narrow vertical strips. If we consider an arbitrary strip with a height of y and a width of dx, we may deduce that dA ( elementary strip’s area) = y dx, here we take y = f. (x). 

It is known as the elementary area, and it may be found at any point inside the region defined by a value of x between a and b.

By adding up the elementary areas of thin strips over the region PQRSP, the region’s total area A between the x-axis, curve y = f(x), and ordinates x = a, x = b, is computed.

As you progress through this chapter, you’ll see how effective these formulae are in identifying these locations. Let’s look at how the chapter unfolds and exposes you to these fantastic applications.

Let’s get into more details on Area Under Simple Curves

If the space under the umbrella is smaller than the combined area of you and your spouse, both of you will get drenched from the sides, however, if the area under the umbrella is larger than the combined area of you and your partner, both of you will be protected from the rain! These kinds of issues fall under the subject of analysis of the area under simple curves, which is what we’ll be talking about right now.

Area Under Curves

Different methods are used to determine the area under the curve, with the antiderivative method being the most prevalent. Knowing the equation of the curve, the borders of the curve, and the axis containing the curve allows you to calculate the area under the curve. There are formulas for finding the areas of conventional figures like squares, rectangles, quadrilaterals, polygons, and circles, but no similar formula exists for finding the area under a curve. The integration procedure aids in the solution of the problem and the determination of the needed area.

Before we get started, it’s important to understand that talking about the area under a curve only makes sense when you have a graph of that curve. Let’s look at an x-function with the formula y = f. (x). The curve shown below is an example of a generic curve whose values can be both positive and negative depending on the value of x.

• The portion of the territory that is entirely above the -axis.
• The area of the curve’s portion is below the -axis.
• The area below the curve where a portion can be found both above and below the -axis.

Geometry may be used to calculate the area under a straight line. Calculating the area beneath a curved line necessitates the use of calculus. The area under a curve is frequently regarded as the total quantity of whatever function is being modeled.

Area Under The Curve –  Between a Curve and A-Line

It is easy to compute the area between two curves by subtracting the area of one curve from that under the line. In this case, the boundary with respect to the axis is the same for both the curve and the line. It is important to note that, in the event that the value of the area of a certain region turns out to be negative, we must take the absolute value of the area and add it to the total of the remaining areas.

Conclusion

Here we learned about the area under simple curves in a detailed manner. The article discusses essential aspects of the topic and explains them with suitable examples. Also, ensure to go through all the relevant questions regarding the topic mentioned in the article for more information. In many cases, two or more curves might be considered to be the boundaries of a region. When an area is encompassed by only two curves, it is possible to calculate it using vertical elements by subtracting the lower function from the higher function and evaluating the integral of the lower function. The area between two curves using horizontal elements can be calculated in a similar way. Subtract the left function from the right function, and you will have the area between two curves.