How to find the area bounded by the curve and xaxis
I'll tell you how to do the second one.
$$y = x^3  3x^2$$
First of all let's find the extrema of the integration by solving those two equations:
1) Set $x = 0$ we find
$$ y = 0 $$
2) Set $y = 0$ we find
$$x^3  3x^2 = 0 ~~~~~ \to ~~~~~ x^2(x + 3) = 0 ~~~~~ \to ~~~~~ x = 3$$
Thence your extrema are $0$ and $3$.
Finally:
$$A = \int_{3}^0 x^3  3x^2\ \text{d}x = \frac{1}{4}x^4  x^3 \bigg_{3}^0 = 0  \left(\frac{(3)^4}{4}  (3)^3\right) = 0 + \frac{81}{4}  27 = 6.75$$
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Road2PreSchool
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Updated on August 16, 2022Comments

Road2PreSchool about 1 year
The text says: Determine the area bounded by the curve and $x$axis. Could somebody solve or at least explain how I should solve these problems.
 $y=x^3+1.5x^2$
 $y=x^33x^2$
 $y=x^2+x+2$ and $y=x^2+x+4$

gambler101 about 7 yearsUse definite integration.

amWhy about 7 yearsThe xaxis is given by $y=0$

Road2PreSchool about 7 yearsyeah I understand that i must use definite integration but how to determine what should i integrate

Road2PreSchool about 7 yearscould it be that the third one result is 8/3

Road2PreSchool about 7 yearsyes thank you but could you explain how can i find the region bounded without a graph ?

Ahmad Bazzi about 7 yearsYou will have to know which part is bounded between the curve and xaxis..Plotting or guessing should be done