   Chapter 4.5, Problem 53E

Chapter
Section
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 estimate:

Area of the region that lies under the given curve using graph and then find the exact area

Explanation

1) Concept:

i) The substitution rule:

If u=g(x) is a differentiable function whose range is I and f is continuous on I, then f(gx)g'xdx=f(u)du. Here,g(x) is substituted as u and then the differentiation is g(x)dx =du

ii) Indefinite integral

xn dx=xn+1n+1+C   (n-1)

2) Given:

y=2x+1 , 0x1

3) Graph:

4) Calculation:

From the graph, the area of the shaded region is split into two parts. One is a square with area 1 and the other is approximately a triangle with area less than  0.4

Therefore, the area of the shaded region from the graph is approximately 1.4.

Now, to find exact area, integrate given function.

A=012x+1dx

Here, use the substitution method.

Substitute 2x+1=u.

Differentiating with respect to x

2dx=du

dx=du2

The limits change and the new limits of integration are calculated by substituting

for x=0, u=20+1=1 and for x=1, u=21+1=3

Therefore, the given integral becomes

=13udu2

Factor out 12

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