Miscellaneous Solids. How do I solve this problem?

Question

Can anybody help me with this

Find the volume in the first octant inside the cylinder $x^2/a^2 +y^2/b^2 =1$ under the plane $z=3x$. Use the given slice in the figure to compute the volume.

Answer 1

Based on the figure, the volume is

$$\int_{y=0}^{b} \frac{xz}{2}dy$$ $$= \int_{y=0}^{b} \frac{3x^2}{2}dy$$ $$= \int_{y=0}^{b} \frac{3}{2}a^2(1 - \frac{y^2}{b^2})dy = a^2b$$

Answer 2

Using ellipse coordenates for the double integral:

$$\int\int_D\left(\int_0^{3x}1\,dz\right)\,dx\,dy=\int\int_D3x\,dx\,dy=3\,a^2\,b\int_0^{\pi/2}\int_0^1\,r^2\,\cos \theta\,dr\,d\theta=\boxed{a^2\,b}$$