SATHEE: Definite Integration Question 21

Definite Integration Question 21

Question 21

If ’ $f$ ’ is a continuous function with $\int_{0}^{x} f (t) d t \to \infty$ as $| x | \to \infty$ , then show that every line $y = m x$ intersects the curve $y^{2} + \int_{0}^{x} f (t) d t = 2$

$(1991, 2 M)$

Show Answer

Solution:

Since, $f$ is continuous function and $\int_{0}^{x} f (t) d t \to \infty$ , as $| x | \to \infty$ . To show that every line $y = m x$ intersects the curve $y^{2} + \int_{0}^{x} f (t) d t = 2$

At $x = 0, y = \pm \sqrt{2}$

Hence, $(0, \sqrt{2}), (0, - \sqrt{2})$ are the point of intersection of the curve with the $Y$ -axis. As $x \to \infty, \int_{0}^{x} f (t) d t \to \infty$ for a particular $x$ (say $x_{n}$ ), then $\int_{0}^{x} f (t) d t = 2$ and for this value of $x, y = 0$

The curve is symmetrical about $X$ -axis.

Thus, we have that there must be some $x$ , such that $f (x_{n}) = 2$ .

Thus, $y = m x$ intersects this closed curve for all values of $m$ .

Play Store

App Store

Ask SATHEE

Welcome to SATHEE !
Select from 'Menu' to explore our services, or ask SATHEE to get started. Let's embark on this journey of growth together! 🌐📚🚀🎓

I'm relatively new and can sometimes make mistakes.
If you notice any error, such as an incorrect solution, please use the thumbs down icon to aid my learning.
To begin your journey now, click on "I understand".

Definite Integration Question 21

Question 21

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu