07_Ex 3.7

Exercise Questions

Questions	Question Links
Q1. Find the area between the $x$ axis and…	3.7 Q-1
Q2. Find the area above the $x$ -axis and…	3.7 Q-2
Q3. Find the area below the curve $\mathrm{y}=3 \sqrt{x}$ and…	3.7 Q-3
Q4. Find the area bounded by $\operatorname{Cos}$ function…	3.7 Q-4
Q5. Find the area between the $x$ -axis and…	3.7 Q-5
Q6. Determine the area bounded by the parabola…	3.7 Q-6
Q7. Find the area bounded by the curve $y=x^{3}+1$ ,…	3.7 Q-7
Q8. Find the area bounded by the curve $y=x^{3}-4 x$ …	3.7 Q-8
Q9. Find the area between the curve $y=x(x-1)(x+1)$ …	3.7 Q-9
Q10. Find the area above the $x$ -axis bounded…	3.7 Q-10
Q11. Find the area between the $x$ -axis and…	3.7 Q-11
Q12. Find the area between the $x$ -axis and…	3.7 Q-12
Q13. Find the area between $x$ -axis and…	3.7 Q-13

Overview

This exercise involves finding the areas under various curves, using integration to calculate the region bounded by each curve and the $x$ -axis. The problems involve quadratic, cubic, trigonometric, and radical functions. Understanding these principles is important for solving real-world problems in physics, engineering, and economics, where determining the area under a curve often represents quantities like distance, work, or total growth.

Key Concepts

Definite Integrals: Used to calculate the area between a curve and the $x$ -axis.
Area under a curve: Represented by the integral of the function over a given interval.
Integration Techniques: Techniques such as power rule, substitution, and using symmetry (like for $\cos(x)$ ) are applied in solving these problems.
Function Behavior: Identifying points of intersection (e.g., where the curve meets the $x$ -axis) helps define the limits of integration.

Important Formulas

For the area under a curve $y = f(x)$ from $x = a$ to $x = b$ , the formula is: $\text{Area} = \int_a^b f(x) , dx$
Power Rule for Integration: $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$
Trigonometric Integrals: $\int \cos(x) , dx = \sin(x)$

Tips and Tricks

When dealing with parabolas, identify the roots or $x$ -intercepts to define the limits of integration.
For trigonometric functions, symmetry can simplify calculations, especially if the curve is periodic.
Break down complex expressions, such as $y = x^3 + 1$ , into factors that are easier to integrate.
In some cases, visualize the curve’s behavior using a table of values, which can clarify the bounds and make integration straightforward.

Summary

This exercise focuses on finding areas under different types of curves using definite integrals. By applying integration techniques and understanding the behavior of each function, you can calculate the enclosed area. It covers curves defined by polynomials, radicals, and trigonometric functions, providing a comprehensive review of how to handle various integral problems.

Reference

By Sir Shazad Sair:

By Great Science Academy: