What is Application of Integrals Class 12: Definition & Uses

Understanding the Definition of Application of Integrals

The Application of Integrals in Class 12 refers to using integral calculus to solve practical problems involving areas, volumes, and other quantities. Integrals help find the accumulated value of a function over an interval.

In simple terms, it answers questions like:

• What is the area under a curve between two points?
• How much volume does a solid formed by rotating a curve have?

This chapter builds on the concept of definite integrals introduced earlier and applies it to geometry and physics problems. NCERT Class 12 Mathematics covers these applications thoroughly to prepare students for board exams.

Calculating Area Under Curves Using Integrals

One of the primary applications of integrals is finding the area under a curve between two points $a$ and $b$. The area $A$ is given by the definite integral:

$$ A = \int_a^b f(x) \, dx $$

Here, $f(x)$ is the function representing the curve.

Key points:

• If the curve lies above the x-axis, the integral gives the exact area.
• If the curve crosses the x-axis, areas below the axis are considered negative, so you may need to split the integral.

#### Example: Find the area under $y = x^2$ from $x=0$ to $x=2$.

$$ A = \int_0^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} \approx 2.67 $$

Finding Volumes of Solids of Revolution

Integrals also help calculate the volume of solids formed by rotating a curve around an axis. This is known as the volume of a solid of revolution.

Formula for volume when rotating about the x-axis:

$$ V = \pi \int_a^b [f(x)]^2 \, dx $$

Where $f(x)$ is the function being rotated.

#### Example: Find the volume generated by rotating the curve $y = x$ from $x=0$ to $x=1$ about the x-axis.

$$ V = \pi \int_0^1 x^2 \, dx = \pi \left[ \frac{x^3}{3} \right]_0^1 = \frac{\pi}{3} $$

This application is crucial in engineering and physics problems.

Applications in Real-Life Problems and Geometry

The application of integrals extends beyond pure mathematics to solve real-world problems:

• Finding areas between curves: When two functions $f(x)$ and $g(x)$ intersect, the area between them is:

$$ A = \int_a^b |f(x) - g(x)| \, dx $$

• Work done by a variable force: Calculated using integrals over the distance.
• Centre of mass and moments: Integrals help find the centroid of irregular shapes.

These applications show how integrals connect abstract math to practical situations.

Comparison: Definite Integral vs Application of Integrals

Understanding the difference between the definite integral concept and its applications is important:

This comparison helps clarify why the application chapter is essential for Class 12 students.

Important Formulas to Remember in Application of Integrals

Here are some key formulas from the Application of Integrals chapter:

• Area under curve $y=f(x)$ from $a$ to $b$:

$$ A = \int_a^b f(x) \, dx $$

• Area between two curves $y=f(x)$ and $y=g(x)$:

$$ A = \int_a^b |f(x) - g(x)| \, dx $$

• Volume of solid of revolution about x-axis:

$$ V = \pi \int_a^b [f(x)]^2 \, dx $$

• Volume of solid of revolution about y-axis:

$$ V = 2\pi \int_a^b x f(x) \, dx $$

Memorising these formulas and understanding their derivation is crucial for exam success.