I am currently trying to do this question but I am having trouble getting the correct answer of 25.
I am also having trouble understanding whether my limits are right and whether my method of doing is correct.
I am trying to take the big regions (with the triangles) - regions of triangle to get the shaded area.
This is my current working$(\int_0^4\int_0^\sqrt xx\:dydx\;-\;\int_1^4\int_0^\sqrt xx\:dydx\;)+\;(\int_0^9\int_{-x}^0x\:dydx\;-\;\int_1^9\int_{-x}^0\:x\:dydx)$
1 Answer
$\begingroup$First of all, to get the area, the integrand should be $1$ or if you like $\mathrm{dA},$ not $x$ (or $x\mathrm{dA}).$ Also, you have the limits of integration incorrect. In the region above the $x-$axis, the limits in the first integral are correct, but the inner limits in the second integral are wrong. $y$ runs from $0$ to the line segment joining joining the points $(1,0)$ and $(4,2)$ You need to use this line to determine the $y$ value.
Below the $x-$ axis you have the limits incorrect in both integrals. In the first integral the lower limit should be $-\sqrt{x}$ not $-x,$ though I suspect this is a typo, and in the second integral, you've made the same sort of error in the inner limits as you did above the $x-$axis.
I think, with these pointers, you'll be able to correct your mistakes. Good luck.
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