   Chapter 4.3, Problem 46E

Chapter
Section
Textbook Problem

# Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y = sec 2 x ,     0 ≤ x ≤ π / 3

To determine

To estimate:

The rough area of the region for given curve then find exact area

Explanation

1) Concept:

i) Fundamental theorem of Calculus, Part 2

If f is continuous on a,b, then

abfxdx=Fb-F(a)

where F is any antiderivative of f, that is, a function F such that F'=f

ii) Secant function rule for antiderivative:

ddxsec2x=tanx

2) Given:

y=sec2x,  0xπ/3

3) Calculation:

The graph of the given curve y=sec2x,  0xπ/3  bounded between x=0 and x=π/3 is

The region enclosed by the curve y=sec2x, x=0, x=π/3 and y=0 is shown by the black region

Here 8 small squares makes up 1 unit square of area. There are about 8.5 squares with in the shaded region. So an estimate of area is 8.5/5  =1.7 sq unit.

To find the exact area of the region enclosed by the given curves,

From above graph, the curve y=fx=sec2x bounded between x=0 and x=π/3, is continuous on 0,π/3, then

By using concept i) (Fundamental theorem of Calculus, Part 2),

Ȣ

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