/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Use the value \(\int_{0}^{1} x^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 77

Use the value \(\int_{0}^{1} x^{2} d x=\frac{1}{3}\) to evaluate each definite integral. Explain your reasoning. (a) \(\int_{-1}^{0} x^{2} d x\) (b) \(\int_{-1}^{1} x^{2} d x\) (c) \(\int_{0}^{1}-x^{2} d x\)

Short Answer

Expert verified
(a) \(\int_{-1}^{0} x^{2} d x = \frac{1}{3}\) \n (b) \(\int_{-1}^{1} x^{2} d x = \frac{2}{3}\) \n (c) \(\int_{0}^{1}-x^{2} d x = -\frac{1}{3}\)

Step by step solution

01

Calculation of \(\int_{-1}^{0} x^{2} d x\)

The integrand is \(x^2\), and it's an even function. This means the integral from -a to a is twice the integral from 0 to a. Since the exercise only asks for the integral from -1 to 0, it would be the same as the integral from 0 to 1, so \(\int_{-1}^{0} x^{2} d x = \frac{1}{3}\).
02

Calculation of \(\int_{-1}^{1} x^{2} d x\)

As stated in Step 1, the integrand \(x^2\) is an even function, so the integral from -1 to 1 is twice the integral from 0 to 1. So, \(\int_{-1}^{1} x^{2} d x = 2 \times \frac{1}{3} = \frac{2}{3}\).
03

Calculation of \(\int_{0}^{1}-x^{2} d x\)

Here the integral is of \(-x^2\) from 0 to 1. We know the integral of \(x^2\) from 0 to 1 is \(\frac{1}{3}\). Then the integral of \(-x^2\) from 0 to 1 would be \(-1 \times \frac{1}{3} = -\frac{1}{3}\).

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

Most popular questions from this chapter

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=y^{2}+1, g(y)=4-2 y $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x e^{-x^{2}}, y=0, x=0, x=1\\\ &\begin{gathered} 51 / 3 \end{gathered} \end{aligned} $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=1-x^{2}, \quad[-1,1] $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{3}^{6} \frac{x}{3 \sqrt{x^{2}-8}} d x $$

State whether the function is even, odd, or neither. $$ g(x)=x^{3}-2 x $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.