   Chapter 13.7, Problem 35E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Find the area below the graph of y   = f ( x ) and above the x-axis for f ( x )   = 24 x e − 3 x . Use the graph of y =   f ( x ) to find the interval for which f ( x )   ≥ 0 and the graph of the integral of f ( x ) over this interval to find the area.

To determine

To calculate: The area below the graph of y=f(x) and above the x axis for function f(x)=24xe3x and find the interval for which f(x)>0 and graph the area for the integral of f(x).

Explanation

Given Information:

The provided function is, f(x)=24xe3x.

Formula used:

The area under the curve f(x) between x=a and x=b is given by,

A=abf(x)dx

The simple power rule for the differentiation is,

ddx(xn)=nxn1

The formula for integration by parts is given by:

udv=uvvdu

Calculation:

Consider the provided function, f(x)=24xe3x

The area under the curve of f(x) and above x axis,

Now, use the formula of integration by parts and split the integrands. Thus,

u=x

Use the simple power rule to differentiate,

du=dx

And,

dv=e3x3dx

Integrate by the rule of exponential integration.

v=e3x3dx=e3x

Now, use the formula of integration by parts,

udv=uvvdu

To obtain the integral as,

80xe3x(3)dx=8[xe3x|0+0e3xdx]=8[limb(be3b+0)+limb(13e3b+13

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