Question

I need need help with this. Find the area of the region enclosed by the lines and curve. x-2=2y^2, x=y+5

Answer 1

You just need to plot and see what you've got. For x-y axes in the usual orientation, you don't have a function with the quadratic unless you restrict for x. It's easier to turn the image on its side and integrate along the y-axis instead.

Doing this, you'll have an element of area as\[{\delta}A={\delta}y(x_{line}-x_{parabola})={\delta}y((y+5)-(2+2y^2))\]that is,\[{\delta}A=(3+y-2y^2){\delta}y \rightarrow A=\int\limits_{-1}^{3/2}3+y-2y^2{dy}\]where the limits of integration have been found for those y-values that yield the same x-values (i.e. points of intersection of the parabola and line). Integrating and subbing in your limits gives\[A=\frac{125}{4}\]

Answer 2

it is nothing but integrating the difference of the two curves ( or curve & a line) & subs the limits which gives the area