To determine To find: The value of the improper integral ∫ − ∞ ∞ x e − x 2 d x . Explanation Definition used: 1. “If f is continuous on ( − ∞ , c ] , then ∫ − ∞ c f ( x ) d x = lim b → − ∞ ∫ b c f ( x ) d x .” 2. “If f is continuous on [ c , ∞ ) , then ∫ c ∞ f ( x ) d x = lim b → ∞ ∫ c b f ( x ) d x .” 3. Let a function f be continuous over the unbounded interval ( − ∞ , ∞ ) . Let c be any real number, and suppose that both the improper integral ∫ − ∞ c f ( x ) d x and ∫ c ∞ f ( x ) d x are convergent. Then the improper integral of f over ( − ∞ , ∞ ) is defined as ∫ − ∞ ∞ f ( x ) d x = ∫ − ∞ c f ( x ) d x + ∫ c ∞ f ( x ) d x , If anyone of the right hand side is divergent then ∫ − ∞ ∞ f ( x ) d x is divergent. Calculation: The improper integral is ∫ − ∞ ∞ x e − x 2 d x . In order to find the above integral, first evaluate the ∫ − ∞ 0 x e − x 2 d x and ∫ 0 ∞ x e − x 2 d x . Evaluate the integral ∫ − ∞ 0 x e − x 2 d x as follows. ∫ − ∞ 0 x e − x 2 d x = lim b → − ∞ ∫ b 0 x e − x 2 d x = − lim b → − ∞ 1 2 ∫ b 0 e − x 2 d ( x 2 ) = − 1 2 lim b → − ∞ [ e − x 2 ] b 0 = − 1 2 lim b → − ∞ ( e − 0 2 ) + 1 2 lim b → − ∞ ( e − b 2 ) On further simplification, ∫ − ∞ 0

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

See similar textbooks

Concept explainers

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 …

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experime…

Videos

Question

Chapter 8.4, Problem 35E

To determine

To find: The value of the improper integral ∫−∞∞xe−x2dx .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.4, Problem 34E

Chapter 8.4, Problem 36E

Students have asked these similar questions

A particle moves along the x-axis for 0 < t < 18 such that its velocity is given by the graph shown below. Find the total distance traveled by the particle during the time interval 4 ≤ t ≤ 8. 8 y 7 6 5 4 32 1 6 7 -1 1 2 3 4 5 -1 -2 -3 -4 56 -6 -8 8 00 Graph of v(t) x 9 10 11 12 13 14 15 16 17 18 19

Using the Chain rule please and thank you

10. [-/3 Points] DETAILS MY NOTES SESSCALCET2 7.2.047. Consider the following. aR- br (a) Set up an integral for the volume a solid torus (the donut-shaped solid shown in the figure) with radii br and aR. (Let a 8 and b = 2.) = dy (b) By interpreting the integral as an area, find the volume V of the torus. V = Need Help? Read It Watch It

Chapter 8 Solutions

Calculus with Applications (11th Edition)

Ch. 8.1 - YOUR TURN 1 Find Ch. 8.1 - YOUR TURN 2 Find Ch. 8.1 - YOUR TURN 3 Find Ch. 8.1 - YOUR TURN 4 Find . Ch. 8.1 - YOUR TURN 5 Find Ch. 8.1 - Prob. 1WE Ch. 8.1 - Find the following. W2. Ch. 8.1 - Find the following. W3. Ch. 8.1 - Find the following. W4. Ch. 8.1 - Use integration by parts to find the integrals in...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.