Find the volume under the surface $z = x^2 + y^2$

Question

I need to find the volume of this surface: $$z = x^2+y^2$$ On the domain limited by: $$y = x^2$$ $$y = 1$$ $$z=0$$ It appears to me that this is the proper way to integrate this: $$\int_{x =-1}^{x=1} \int_{y=0}^{y= x^2}(x^2+y^2)dydx$$ However, when I do this I get the result $V = 52/105$ and the correct answer is supposed to be $V= \frac{88}{105}$. Did I use the wrong limits for integration?

Answer 1

Yes, you used the wrong limits. It should have been:$$\int_{-1}^1\int_{x^2}^1x^2+y^2\,\mathrm dy\,\mathrm dx.$$You will get $\frac{88}{105}$ indeed.