   Chapter 5, Problem 21TYS ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# In Exercises 21 and 22, use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f and the x-axis over the interval. Compare your result with the exact area obtained using a definite integral. Sketch the region. f ( x ) = 3 x 2 , [ 0 , 1 ]

To determine

To calculate: The area of the region bounded by the graph of the function f(x)=3x2 with x-axis by the midpoint rule with n as 4 and also calculate the exact area by the use of a definite integral and sketch the region.

Explanation

Given Information:

The provided function is f(x)=3x2 and the intervals is intervals are [0,1]

Formula used:

Steps to solve a definite integral abf(x)dx with the help of midpoint rule.

Step 1: For a given interval [a,b] divide it into n subintervals with having width,

Δx=ban

Step 2: Evaluate the midpoint for the given subinterval. Midpoints={x1,x2,x3,xn}

Step 3: Find the value of f at each midpoint and make the sum as shown below,

abf(x)dxban[f(x1)+f(x2)+f(x3)++f(xn)]

Steps to solve a definite integral abf(x)dx with the help of basic integration rule is,

abkf(x)dx=kabf(x)dx

Where k is a constant.

Calculation:

Consider the function,

f(x)=3x2

The intervals are [0,1] with n=4

Now divide the provided interval into 4 subparts as shown below,

Δx=104=14

Therefore, the 4 subintervals are, [0,14],[14,12],[12,34] and [34,1].

Now find the mid points of these intervals because each subinterval has a width of 14.

Therefore, the mid points of these interval are 18,38,58 and 78.

Here the mid points 18,38,58 and 78 lies in the middle of [0,14],[14,12],[12,34],[34,1] intervals respectively

### 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

Mathematical Applications for the Management, Life, and Social Sciences

#### Evaluate the integral. 21. dt(t21)2

Single Variable Calculus: Early Transcendentals

#### limx0+(1+x)1/x= a) e b) e c) 1e d) 1e

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