   Chapter 4.2, Problem 49E

Chapter
Section
Textbook Problem

Finding Area by the Limit Definition In Exercises 47-56, use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region. y = x 2 + 2 , ⌈ 0 , 1 ⌉

To determine

To calculate: Area of the region bounded by y=x2+2, in the interval [0,1] and the x-axis.

Explanation

Given: y=x2+2, in the interval [0,1].

Formula used: Formula for the sum of squares of first n natural:

i=1ni2=n(n+1)(2n+1)6

The sum of a constant n times is written as:i=1nc=nc

Use right endpoints area is written as:Area=limni=1ny(Mi)(Δx), where, Mi are the right endpoints.

Calculation: Function y is continuous and non-negative in the interval [0,1].

Partition the interval into n subintervals each of width, Δx:

Δx=10n=1n

Area can be calculated by left endpoints (mi) or right endpoints (Mi).

Right endpoints (Mi) are:

0+in=in,

i=1,2,3,.......,n

Area=limni=1ny(Mi)(Δx) where, Mi are the right endpoints, and Mi=in.

Use value of y(in),

Area=limni=1n((in)2+2)(1n)=limni=1n1n3(i2)+limni=1n(2n)=lim</

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Simplify the expressions in Exercises 97106. x1/2y2x1/2y

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the derivative. Simplify where possible. 37. y = ecosh 3x

Single Variable Calculus: Early Transcendentals, Volume I

Find each product. x(3x22x+5)

Elementary Technical Mathematics

Given: l1l2l3l4,AB=7,BC=5,CD=4,EF=6 Find: FG,GH,EH

Elementary Geometry For College Students, 7e

If f(x) = sin 2x, an upper bound for |f(n + 1)(x)| is 2 2n 2n + 1 22n

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

The point P graphed at the right has spherical coordinates:

Study Guide for Stewart's Multivariable Calculus, 8th

Explainthe FOIL method.

College Algebra (MindTap Course List)

Why are studies that examine the effects of aging not considered true experiments?

Research Methods for the Behavioral Sciences (MindTap Course List) 