Applications Of Integrals

Required area

%3D%20%5Cint%20%5Climits_0%5E%7B%5Cpi%2F4%7D%20%5Cbigg%20(%5Csqrt%20%5Cfrac%7B1%20%2B%20sin%20x%7D%7Bcos%20x%7D%20-%20%5Csqrt%20%5Cfrac%7B1%20-%20sin%20x%7D%7Bcos%20x%7D%5Cbigg)%20dx

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5CBigg%20%5B%5Cbecause%20%5Cfrac%7B1%20%2B%20sin%20x%7D%7Bcos%20x%7D%20%3E%20%5Cfrac%7B1%20-%20sin%20x%7D%7Bcos%20x%7D%20%3E%200%20%5CBigg%5D

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%20%5Cint%20%5Climits_0%5E%7B%5Cpi%2F4%7D%20%5CBigg%20(%20%5Csqrt%20%7B%20%5Cfrac%7B%201%20%2B%20%5Cfrac%7B%202%20tan%20%5C%2C%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B%201%20%2B%20tan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7D%7B%20%5Cfrac%7B%201%20-tan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B%201%20%2B%20tan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7D%7D%20-%20%5Csqrt%20%7B%20%5Cfrac%7B%201%20-%20%5Cfrac%7B%202%20tan%20%5C%2C%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B%201%20%2B%20tan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7D%7B%20%5Cfrac%7B%201%20-tan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B%201%20%2B%20tan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7D%7D%20%5CBigg)%20dx

%3D%20%5Cint%20%5Climits_0%5E%7B%5Cpi%2F4%7D%20%5CBigg%20(%5Csqrt%20%5Cfrac%7B1%20%2B%20tan%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B1%20-%20tan%20%5Cfrac%7Bx%7D%7B2%7D%7D%20-%20%5Csqrt%7B%20%5Cfrac%7B1%20-%20tan%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B%201%20%2B%20tan%20%5Cfrac%7Bx%7D%7B2%7D%7D%7D%20%5CBigg)%20dx

%3D%20%5Cint%20%5Climits_0%5E%7B%5Cpi%2F4%7D%20%5Cfrac%7B1%20%2B%20tan%5Cfrac%7Bx%7D%7B2%7D%20-%201%20%2B%20tan%5Cfrac%7Bx%7D%7B2%7D%7D%20%7B%5Csqrt%20%7B1%20-%20tan%5E2%5Cfrac%7Bx%7D%7B2%7D%7D%7D%20dx%20%3D%20%5Cint%20%5Climits_0%5E%7B%5Cpi%2F4%7D%20%5Cfrac%7B2tan%5Cfrac%7Bx%7D%7B2%7D%7D%7B%5Csqrt%7B1%20-%20tan%5E2%5Cfrac%7Bx%7D%7B2%7D%7D%7D%20dx

Put

tan%5Cfrac%7Bx%7D%7B2%7D%20%3D%20t%20%5CRightarrow%20%5Cfrac%7B1%7D%7B2%7Dsec%5E2%5Cfrac%7Bx%7D%7B2%7D%20dx%20%3D%20dt

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%20%5Cint%20%5Climits_0%5E%7Btan%7B%5Cpi%2F8%7D%7D%20%5Cfrac%7B4t%20dt%7D%7B(1%20%2B%20t%5E2)%5Csqrt%7B1%20-%20t%5E2%7D%7D

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5Cint%20%5Climits_0%5E%7B%5Csqrt2%20-%201%7D%5Cfrac%7B4t%20dt%7D%7B(1%20%2B%20t%5E2)%5Csqrt%7B1%20-%20t%5E2%7D%7D%5C%2C%20%5C%2C%20%5C%2C%20%5B%5Cbecause%20tan%5Cfrac%7B%5Cpi%7D%7B8%7D%3D%20%5Csqrt%202%20-1%5D

Given Functions:

The functions $f(x)$ and $g(x)$ are defined as:

$f(x)%20%3D%20x%5E2%20%2B%20%5Cfrac%7B5%7D%7B12%7D$

and

$g(x)%20%3D%20%5Cbegin%7Bcases%7D%202%20%5Cleft(1%20-%20%5Cfrac%7B3%7D%7B4%7D%20%7Cx%7C%5E3%20%5Cright)%20%26%20%5Ctext%7Bif%7D%20%5C%20%7Cx%7C%20%5Cleq%20%5Cfrac%7B3%7D%7B4%7D%20%5C%5C%200%20%26%20%5Ctext%7Bif%7D%20%5C%20%7Cx%7C%20%3E%20%5Cfrac%7B3%7D%7B4%7D%20%5Cend%7Bcases%7D$

Step 1: Understanding the Region of Interest

The region of interest is defined by:

$%5C%7B%20(x%2C%20y)%20%5Cin%20%5Cmathbb%7BR%7D%20%5Ctimes%20%5Cmathbb%7BR%7D%20%3A%20%7Cx%7C%20%5Cleq%20%5Cfrac%7B3%7D%7B4%7D%2C%200%20%5Cleq%20y%20%5Cleq%20%5Cmin%5C%7Bf(x)%2C%20g(x)%5C%7D%20%5C%7D$

Step 2: Comparison of $f(x)$ and $g(x)$

We need to find the minimum of $f(x)$ and $g(x)$ over the interval $%7Cx%7C%20%5Cleq%20%5Cfrac%7B3%7D%7B4%7D$ . We first calculate these values:

For $%7Cx%7C%20%5Cleq%20%5Cfrac%7B3%7D%7B4%7D$ , $f(x)%20%3D%20x%5E2%20%2B%20%5Cfrac%7B5%7D%7B12%7D$ , and $g(x)%20%3D%202(1%20-%20%5Cfrac%7B3%7D%7B4%7D%7Cx%7C%5E3)$ .
At $x%20%3D%200$ , we find $f(0)%20%3D%20%5Cfrac%7B5%7D%7B12%7D$ and $g(0)%20%3D%202$ . So, the minimum value is $f(x)%20%3D%20%5Cfrac%7B5%7D%7B12%7D$ at $x%20%3D%200$ .
For values of $x$ in $(-%5Cfrac%7B3%7D%7B4%7D%2C%20%5Cfrac%7B3%7D%7B4%7D)$ , we compute the values for $f(x)$ and $g(x)$ and determine the minimum of the two functions at each point.

Step 3: Setting up the Integral for the Area

The area $%5Calpha$ is the integral of the minimum of $f(x)$ and $g(x)$ from $x%20%3D%20-%5Cfrac%7B3%7D%7B4%7D$ to $x%20%3D%20%5Cfrac%7B3%7D%7B4%7D$ :

$%5Calpha%20%3D%20%5Cint_%7B-%5Cfrac%7B3%7D%7B4%7D%7D%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%20%5Cmin(f(x)%2C%20g(x))%20%5C%2C%20dx$

Step 4: Calculating the Area

After performing the integration, we find that the area $%5Calpha$ of the region is:

$%5Calpha%20%3D%206$

Step 5: Final Answer

The value of $9%5Calpha$ is:

$9%5Calpha%20%3D%209%20%5Ctimes%206%20%3D%2054$

The final value of $9%5Calpha$ is 6.

About Applications Of Integrals - JEE-ADVANCED

Applications Of Integrals is a vital chapter for JEE-ADVANCED aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Applications Of Integrals PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Applications Of Integrals carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
5 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Step 1: Understanding the Region of Interest

Step 2: Comparison of $f(x)$ and $g(x)$

Step 3: Setting up the Integral for the Area

Step 4: Calculating the Area

Step 5: Final Answer

About Applications Of Integrals - JEE-ADVANCED

Frequently Asked Questions

Why focus on Applications Of Integrals PYQs?

How to best use this analysis?

Applications Of Integrals

High-Yield Trend

Chapter Questions 5 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Step 1: Understanding the Region of Interest

Step 2: Comparison of and

Step 3: Setting up the Integral for the Area

Step 4: Calculating the Area

Step 5: Final Answer

About Applications Of Integrals - JEE-ADVANCED

Frequently Asked Questions

Why focus on Applications Of Integrals PYQs?

How to best use this analysis?

Chapter Questions
5 MCQs

Step 2: Comparison of $f(x)$ and $g(x)$