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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Let be the region bounded by the curves y = tan ( x 2 ) , x = 1 , and y = 0 . Use the Midpoint Rule with n = 4 to estimate the following quantities.

(a) The area of

(b) The volume obtained by rotating about the x -axis

To determine

a)

To estimate:

The area of R

Explanation

1) Concept:

Midpoint rule:

abfxdx=i=0nf(xi-)x

where

 x=b-an&xi-=12xi-1+xi

midpoint of xi-1,xi. n  is the number of subintervals.

2) Given:

 y=tanx2, x=1 & y=0 with n=4

3) Calculation:

The bounded region by the given curve is as below:

The equation of the curve is

y=tanx2& y=0

tanx2=0  only when x2=0 i.e. x=0  and rest of the zeros are beyond the bounded region.

Use midpoint rule on 0,1 for fx=tanx2.

With n=4, a=0 & b=1, the interval width is

x=1-04=14

The 4  subintervals are

0, 14 ,14, 12,12, 34,34 ,1

Midpoints of subintervals are

x1-=12x0+x1=120+14=18

x2-=12x1+x2=1214+12=1268=1234=38

x3

To determine

b)

To estimate:

The volume obtained by rotating R  about x  axis.

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