How do you find the area of the region bounded by the given curves #y = 6x^2lnx # and #y = 24lnx#?

Answer 1

#int_1^2(24lnx-6x^2lnx)dx#
#=32ln2 - 58/3#

First set the equations equal to each other, and find the intersection points, then find the area between the curves.

#color(red)(y=6x^2lnx)#
#color(blue)(y=24lnx)#

#6x^2lnx=24lnx#
#0=24lnx-6x^2lnx#
#0=(-6)(lnx)(x^2-4)#
#x=1,2#

By setting the equations equal to each other, I found that the functions intersect at #x=1# and #x=2#:

#"Area"=int_1^2(24lnx-6x^2lnx)dx #

Focus on the unbounded (indefinite) integral to solve:
#int(24lnx-6x^2lnx)dx#
#=-6 int (lnx)(x^2-4) dx#

Use integration by parts: #int color(blue)(u) color(green)(dv)= vu - int v du#
#((u=color(blue)(lnx)),(du=1/x dx))((v=1/3x^3-4x),(dv=color(green)(x^2-4)))#

#=-6[(lnx)(1/3x^3-4x)- int(1/x)(1/3x^3-4x)dx]#

#=-6[(lnx)(1/3x^2-4x)-int(1/3x^2-4)dx]#

#=-6[(lnx)(1/3x^2-4x)-(1/9x^3-4x)]#

#=-6(lnx)(1/3x^3-4x)+2/3x^3-24x#

Now go back to the definite integral:
#int_1^2(24lnx-6x^2lnx)dx#
#=[-6(lnx)(1/3x^3-4x)+2/3x^3-24x]_1^2#

#=-6ln2(8/3-8) + 16/3 - 48 - 2/3 + 24#

#= 32ln2- 58/3#

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Answer 2

To find the area of the region bounded by the given curves y = 6x^2lnx and y = 24lnx, you need to first determine the points of intersection of these curves. Then, integrate the absolute difference of the two functions between these points of intersection. The points of intersection can be found by solving the equation 6x^2lnx = 24lnx. After finding the points of intersection, integrate |6x^2lnx - 24lnx| with respect to x over the appropriate interval.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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