Explain area under the curve jee

Calculation of area is one of important topic in integral calculus. The area under the curve is a change in the quantity with respect to time . On integrating it ,we can find the amount of quantity changed.

The area under the curve is a fundamental concept in integral calculus that quantities the total area between the graph of functions and the x- axis over a specified interval.

If f(x) is a continuous function defined on the interval [a,b] ,the area under the curve from x=a to x=b is given by definite integral.

For more information,you can check Application of integrals .

Area under the Curve - One of the most relevant concepts in integral calculus. The area inside a curve, x-axis, and two vertical lines is called the area under the curve. It is used in science and engineering in physics, mathematics, economics, among many others.

For JEE students, Area under Curve forms an integral part in solving problems related to definite integration on certain topics.

Calculating definite integral of any function over a specific range yields area under curve.

Method to calculate definite integral: It involves making use of the Fundamental Theorem of Calculus.

Kinematics:

Displacement is equivalent to the area under the velocity-time graph.

Change in velocity is equivalent to the area under the acceleration-time graph.

Work Done by a Variable Force:

Area under the force-displacement curve denotes work done by any variable force.

Key Formulas and Techniques:

Definite Integral:

∫[a,b] f(x) dx

This integral calculates the area under the curve y = f(x) between x = a and x = b.

Area Between Curves:

Consider two curves y = f(x) and y = g(x). The area between them in the interval from x = a to x = b is computed by the formula:

∫[a,b] |f(x) - g(x)| dx

Area Bounded by Curves and Axes:

To find the area bounded by a curve, the x-axis, and two vertical lines, we integrate the absolute value of the function between the given limits.

Guidelines on Area Under the Curve Problems

Sketch the Graph: Drawing the graph is essential in determining the limits of integration and the appropriate formula that should be applied.

Identify the Curves: Clearly identify the curves bounding the region.

Identify the Limits of Integration: The x-coordinates of the points of intersection of the curves should be found.

Setup the Integral: Write the definite integral based on the region and the functions involved.

Evaluate the Integral: Employ some integration techniques to evaluate the definite integral.

By knowing what area under the curve is and by mastering the art of integration, all sorts of problems in JEE can be effectively solved. https://engineering.careers360.com/exams/jee-main/area-between-two-curves-calculus-practice-question-mcq