Question #dc81f

1 Answer
Radioactive Teddybear
Nov 22, 2017

#5(1/(2e^2)+7)# or #~35.33834#

Explanation:

let's make the exponential decay graph #f(x)# graph{e^(-0.4x) [-7.024, 7.024, -3.51, 3.514]}
and the linear graph #g(x)#
graph{2.4x+1 [-7.024, 7.024, -3.51, 3.514]}

Judging by the graphs, the linear graph will be the "upper curve" from #0# to #5# since it is greater than the exponential decay graph at all points #(0,5]#

So it will be integration of upper area minus lower area for the domain, or #int_0^5g(x)-f(x)dx#.
Then you substitute the equations, so
#int_0^5(2.4x+1)-(e^(-0.4x))dx#
=#int_0^5 2.4x+1-e^(-0.4x)dx#

Then you can integrate and solve using the 2nd Fundamental Theorem of Calculus.

#int_0^5 2.4x+1-e^(-0.4x)dx#
=#(1.2x^2+x+2.5e^(-0.4x))|_0^5#
=#1.2*5^2+5+2.5e^(-0.4*5)-(0)#
=#5(1/(2e^2)+7)# or #~35.33834#

Ask if you need any clarification on this. Thanks for asking though.

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