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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.

f ( x ) = x 2 + 1 + 2 x , 4 x 7

To determine

To find: The expression for area under the graph.

Explanation

Given information:

The function is f(x)=x2+1+2x.

The upper limit is b=7 and lower limit is a=4.

State the definition 2 as shown below:

A=limnRn=limn[f(x1)Δx+f(x2)Δx+...+f(xn)Δx]=limni=1nf(xi)Δx (1)

The area A of the region (S) under the graph f of a continuous function is the sum of area of the approximating rectangles.

Calculation:

Find the width of the interval (Δx) using the relation:

Δx=ban (2)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute 7 for b and 4 for a in Equation (2).

Δx=74n=3n

Find the value of xi using the relation:

xi=a+iΔx (3)

Substitute 4 for a and 3n for Δx in Equation (3).

xi=4+i3n=4+3in

Use definition 2 to obtain the expression for area under the curve f

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