What is the area between the graph of y=x^3 and the axis from x=3 to x=4?

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2 Answers
turksvids
Jan 1, 2018

#175/4#

Explanation:

We're trying to find #int_3^4x^3dx#. We'll use the Fundamental Theorem to evaluate the integral:

Since #intx^3dx = 1/4x^4+C#, say #f(x)=1/4x^4#. (The #+C# won't matter for definite integrals, so don't worry about its value.)

#int_3^4x^3dx = f(4)-f(3) = 1/4(4)^4-1/4(3)^4#
#=64-81/4=(256-81)/4 = 175/4#

Jim G.
Jan 1, 2018

#175/4to(B)#

Explanation:

#"the area under a curve is found using"#

#•color(white)(x)"area(A) "=int_a^bydx#

#"here "y=x^3" and "a=3,b=4#

#rArrA=int_3^4x^3dx#

#"integrate using the "color(blue)"power rule"#

#•color(white)(x)int(ax^n)dx=a/(n+1)x^(n+1)to(n!=-1)#

#rArrint_3^4x^3dx=[1/4x^4]_3^4#

#"to evaluate consider"#

#intf(x)dx=F(x)+c#

#rArrint_a^bf(x)=(F(b)+c]-[F(a)+c]=F(b)-F(a)#

#"that is evaluate at x =b and x = a and subtract"#

#rArr[1/4x^4]_3^4#

#=(1/4xx4^4)-(1/4xx3^4)#

#=64-81/4=256/4-81/4=175/4to(B)#

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