Q. 1 TF Functions defined by area accumu... [FREE SOLUTION]

Chapter 4: Q. 1 TF (page 354)

Functions defined by area accumulation: We know that for fixed real numbers a and b and an integrable functionf, the definite integral ${鈭�}_{a}^{b} f (x) d x$ is a real number. For different real values of b, we get (potentially) different values for the integral ${鈭�}_{a}^{b} f (x) d x .$
Make a table of the values of the integral ${鈭�}_{0}^{b} 2 x d x$ corresponding to the values $- 3, - 2, - 1, 0, 1, 2, and 3$ for b. Conjecture a formula for the relationship between the values of b and the corresponding value of the integral.

Short Answer

Expert verified

The formula for the integral is ${鈭�}_{0}^{b} 2 x d x = b^{2}$ and the table of values of integral for different values of b is following.

Step by step solution

Step 1. Given information.

The given integral is ${鈭�}_{a}^{b} f (x) d x = {鈭�}_{0}^{b} 2 x d x .$

Given values of b are $- 3, - 2, - 1, 0, 1, 2, and 3 .$

Step 2. Integral of ∫0b2x dx.

Determine the integral of ${鈭�}_{0}^{b} 2 x d x .$

localid="1648629126228" ${鈭�}_{a}^{b} f (x) d x = {鈭�}_{0}^{b} 2 x d x = 2 {鈭�}_{0}^{b} x d x = 2 {[\frac{x^{2}}{2}]}_{0}^{b} = 2 \frac{b^{2}}{2} = b^{2}$

Solocalid="1648628956509" ${鈭�}_{0}^{b} 2 x d x = b^{2} .$

Step 2. Values of integral.

Substitute $- 3, - 2, - 1, 0, 1, 2, and 3$ for bin the integral.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Integral of ∫0b2x dx.

Step 2. Values of integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.