   Chapter 4.3, Problem 45E

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 = sin x ,     0 ≤ x ≤ π

To determine

To estimate:

The rough area of the region for the given curve and 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) Cosine function rule for antiderivative:

ddxcosx=-sinx

2) Given:

y=sinx,  0xπ

3) Calculation:

The curves y=sinx,  0xπ are given by,

From above graph, the given curve y=sinx,  0xπ bounded between x=0 and x=π

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

Here 50 small squares constitute 1 sq unit area. There are about 93 small squares within the  region. So a rough estimate of area is 93/50 = 1.86 sq unit

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