What is Application of Integrals Class 12: Definition & Uses

Understanding the Definition of Application of Integrals

The Application of Integrals in Class 12 Mathematics refers to using definite integrals to solve practical problems such as finding areas, volumes, and lengths. Integrals help calculate quantities that are not easily measured by simple formulas. In NCERT, this chapter introduces how integrals can be applied beyond theoretical calculus, making it a valuable tool for various fields.

Key points:

• Integrals sum infinitely small quantities to find total value.
• Definite integrals evaluate the area under a curve between two limits.
• Applications include areas between curves, volumes of solids, and more.

This chapter builds on your knowledge of integration from Class 11 and extends it to real-world applications.

Finding Area Under Curves Using Integrals

One of the primary applications of integrals is to find the area bounded by curves and the x-axis or between two curves.

Formula for Area Between Curve and X-axis

For a continuous function $f(x)$ on $[a, b]$, the area under the curve is:

$$ ext{Area} = \int_a^b |f(x)| \, dx$$

If $f(x)$ is positive in $[a,b]$, then:

$$ ext{Area} = \int_a^b f(x) \, dx$$

Area Between Two Curves

If $f(x)$ and $g(x)$ are two functions such that $f(x) \geq g(x)$ on $[a,b]$, then:

$$ ext{Area} = \int_a^b [f(x) - g(x)] \, dx$$

#### Worked Example: Find the area bounded by $y = x^2$ and $y = x + 2$.

• Find points of intersection by equating: $x^2 = x + 2 \Rightarrow x^2 - x - 2 = 0$
• Roots: $x = 2$ and $x = -1$
• Area:

$$\int_{-1}^2 [(x + 2) - x^2] \, dx = \left[ \frac{x^2}{2} + 2x - \frac{x^3}{3} \right]_{-1}^2 = \frac{9}{2}$$

Calculating Volume of Solids of Revolution

Application of integrals also includes finding the volume of solids formed by rotating a curve around an axis, called solids of revolution.

Volume About X-axis

If the curve $y = f(x)$ is rotated about the x-axis from $x=a$ to $x=b$, the volume is:

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

Volume About Y-axis

If the curve $x = g(y)$ is rotated about the y-axis from $y=c$ to $y=d$, the volume is:

$$V = \pi \int_c^d [g(y)]^2 \, dy$$

#### Worked Example: Find the volume of the solid formed by rotating the curve $y = \sqrt{x}$ from $x=0$ to $x=4$ about the x-axis.

$$V = \pi \int_0^4 (\sqrt{x})^2 \, dx = \pi \int_0^4 x \, dx = \pi \left[ \frac{x^2}{2} \right]_0^4 = \pi \times \frac{16}{2} = 8\pi$$

Comparing Area and Volume Formulas in Application of Integrals

Understanding the difference between area and volume calculations using integrals is essential.

This table helps students quickly identify the correct integral formula for different applications.

Other Important Applications in Class 12 NCERT

Besides area and volume, integrals are used in other applications such as:

• Length of a curve: The length $L$ of a curve $y=f(x)$ from $x=a$ to $x=b$ is:

$$L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$$

• Average value of a function:

$$f_{avg} = \frac{1}{b - a} \int_a^b f(x) \, dx$$

• Work done by a variable force: If force $F(x)$ varies with position $x$, work done moving from $a$ to $b$ is:

$$W = \int_a^b F(x) \, dx$$

These applications show the versatility of integrals in solving physical and mathematical problems.

Tips for Solving Application of Integrals Problems in Exams

To excel in Application of Integrals questions in Class 12 NCERT exams, keep these tips in mind:

• Understand the problem: Identify whether you need area, volume, length, or work.
• Sketch the graph: Visualizing curves helps set correct limits and functions.
• Set correct limits: Determine intersection points or given interval clearly.
• Choose the right formula: Refer to the comparison table for formulas.
• Simplify integrand: Algebraic simplification before integration saves time.
• Practice problems: Regular practice improves speed and accuracy.

By following these steps, students can confidently tackle application-based integral problems.