   Chapter 5.5, Problem 75E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. y = 2 x + 1 , 0   ≤ x ≤   1

To determine

To find: area of the region that lies under the given curve using a graph.

To find: the exact area for the function y=2x+1,0x1.

Explanation

Given:

The function is y=2x+1,0x1.

The region lies between 0 to 1.

Calculation:

Show theequation as below:

y=2x+1 (1)

Plot a graph for the equation y=2x+1,0x1 using the calculation as follows:

Calculate y value using Equation (1).

Substitute 0 for x in Equation (1).

y=2x+1=(2×0)+1=1=1

Hence, the co-ordinate of (x,y) is (0,1).

Calculate y value using Equation (1).

Substitute 1 for x in Equation (1).

y=2x+1=(2×1)+1=3=1.732

Hence, the co-ordinate of (x,y) is (1,1.7)

Draw the graph for the equation y=2x+1 using calculated (x,y) co-ordinates as in Figure 1.

Refer to Figure (1).

Take the curved region (1) as a triangle.

Calculate the area of shaded region.

A=(b1h1)+( 12b2h2)

Substitute 1 for b1, 1 for h1, 1 for b2, and 0.7 for h2.

A=(b1h1)+( 12b2h2)=(1×1)+( 12×1×0.7)=1+0.35=1.35

Hence, the area of the region is 1.35_.

Show the integral function as below:

A=01f(x)dx (2)

Substitute (2x+1) for f(x) in Equation (2)

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