Question

answer if you know
this is calculus so don't put idk or whatever
show all work and resons


the reigon R is bounded by the x axis,the y axis and y=√x and y=6-x
the area of the reigon is 22/3
the reigon R is the base of a solid. for every y, where 0≤y≤2, the cross section of the solid taken perpendicular to the y axis is a rectangle whose base lies in R and whose heigh is is 2y. write, but do not evaluate, and integral expression that gives the volume of the solid

Answer 1

I've attached a plot of one such cross-section (orange) over the region in the x-y plane (blue), including the bounding curves (red). (I've set $y=1.25$ for this example.)

The length of each cross section (the side lying in the base) has length determined by the horizontal distance $x$ between the y-axis $x=0$ and the curve $y=\sqrt x$. In terms of $y$, this distance is $x=y^2$. The height of each cross section is twice the value of $y$, so the area of each rectangular cross section should be $2y^3$.

This means the volume would be given by the integral

$\displaystyle\int_0^22y^3\,\mathrm dy$