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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Use a graph to estimate the x-intercepts of the curve y = 1 2 x 5 x 4 . Then use this information to estimate the area of the region that lies under the curve and above the x-axis.

Repeat Exercise 43 for the curve y = 2 x + 3 x 4 2 x 6 .

To determine

To estimate:

The x-intercepts and using the intercepts, find the area of the region that lies under the curve and above the  x  axis.

Explanation

1) Concept:

The fundamental theorem of calculus part 2: If  f  is continuous on a,b, then abfxdx=Fb-Fa,  where  F  is any antiderivative of f , that is, the function  F  such that F'=f .

2) Formula:

fx+gxdx=fxdx+gxdx

xndx=xn+1n+1+C

kdx=kx+C

3) Given:

y=2x+3x4-2x6

4) Calculation:

Consider the given curve,

y=2x+3x4-2x6

The graph of the given curve is

By using the graph of given curve, the x  intercept of the given curve is x0, x1.38 

Next, find the area of the region that lies under the curve and above the  x  axis.

From the graph, the curve  y=2x+3x4-2x6 bounded between x=0  and  x=1.38  is continuous on 0, 1.38 and then, the area of the region that lies under the curve and above the x  axis is

01.38(2x+3x4-2x6)dx

To find area of region lies under the curve and above the x  axis, evaluate the definite integral

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