Integral Calculus: Overview, Questions, Preparation

Calculus is among the most recent, and most complex branches of mathematics. The two parts of calculus are differentiation and integration. Here is a brief overview of what integration is and what its implications are.

What is Integration?

Put very simply; integration is the summation of infinitesimal elements of a function. The integral of a function is also known as its antiderivative. This is because for a function f with derivative f’, the integral of f’ is f. Similarly, for a function g with integral g”, the derivative of g” is g. Integrals are not just restricted to the world of mathematics. Integrals are used widely in physics, chemistry, engineering, and a number of other domains.

Integrals are generally used for two broad purposes. The first purpose is to find the function when the derivative of the function has been provided. The second is to find the area bounded by the curve of a function between two points, or the general formula of the area bounded by a function.

Types of Integrals

There are two types of integrals. The first among these is a definite integral. The definite integral is more widely used in real life and has more tangible physical implications. A definite integral is an integral which is bounded by two points. It has an upper limit and a lower limit, and the integral is calculated between these limits. On a Cartesian plane, this can be expressed as the area bounded by a curve between two points on an axis. The resultant of a definite integral is generally not a formula, but a constant number, unless one of the limits is a variable.

The second type of integral is known as an indefinite integral. An indefinite integral does not have an upper limit or a lower limit. It cannot be expressed completely on a Cartesian plane and provides the general function of the area bounded by a curve. The resultant for an indefinite integral is generally a function in terms of an independent variable, followed by an arbitrary constant. A definite integral is basically the difference among the values of an indefinite integral at the upper and lower limit.

Importance and Weightage

The topic of integral calculus contains a large number of formulae that need to be memorized by the student. In order to facilitate this, a number of examples are required to be completed in class, along with explanations.

From the examination point of view, integral calculus can be up to 20 marks, forming 1, 2, 4, and 6 mark questions. It generally forms a significant part of the syllabus of Class XII.

1. Calculate the integral ₀∫⁵⁰(x²+4x+2)dx

Solution.

₀∫⁵⁰(x²+4x+2)dx
=₀∫⁵⁰(x²)dx+₀∫⁵⁰4xdx+₀∫⁵⁰2dx

=(50³-0³)/3 + 2(50²-0²)+2(50-0)

= 125000/3 + 5000 + 100

= 46,766.66

2. Calculate the integral ∫ (x² +sinx)dx

Solution.

∫ (x² +sinx)dx

= ∫ (x²)dx+(sinx)dx

= x³/3 - cosx + c

3. Calculate the integral

Solution. 

∫x/√1-x²dx

Let x=sinx

dx=cosxdx

∫x/√1-x²dx = ∫(sinx/cosx)cosxdx

=∫ sinx dx

=cosx+c

=√1-x²+c

Q: How can a definite integral be expressed on the Cartesian plane?

Q: What is a definite integral in terms of the corresponding indefinite integral?

Q: What are the two types of integrals?

Q: What is an arbitrary constant?

Q: Why is an integral called an anti-derivative?