Answer to Question #129613 in Calculus for Jade

Question #129613
Find the area between the x-axis and the graph of f(x)=x^3-1 from x= 0 to x=4
1
Expert's answer
2020-08-23T17:08:32-0400

As per the question,

Let y=f(x)=x"^3"-1

at x=0,y=-1

also

at y=0,x=1

The given curve passes through(0,-1),(1,0)

Therefore the area under these curve is divided into two parts

i.e. from x=0 to x=1 and from x=1 to x=4

Area= "\\mid" "\\intop_{0}^{1}" f(x)dx"\\mid" +"\\intop_{1}^{4}" f(x)dx

="\\mid\\intop_{0}^{1}" (x"^3" -1)dx"\\mid" + "\\intop_{1}^{4}" (x"^3-" 1)dx


= "\\mid" "\\frac{x^4}{4}-x\\mid_{0}^{1} + \\mid\\frac{x^4}{4}-x\\mid_1^{4}"

Putting the upper and lower limits we get

="\\mid\\frac{1}{4}-1\\mid" +"(\\frac{256}{4}-4)-(\\frac{1}{4}-1)"

="\\frac{3}{4}+\\frac{3}{4}+60"

=0.75+0.75+60

=61.5 square units


Hence the area under curve is 61.5 sq. units




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