1. integral calculus

find the area bounded by the curve y^2=2x+3 and y=x

2. Here's the graph. You should be able to find your limits from it.

3. Hello, alderon!

Find the area bounded by the curves: .$\displaystyle \begin{array}{cccc}y^2 & = & 2x+3 & {\color{blue}[1]} \\ y & = & x & {\color{blue}[2]}\end{array}$
We'll do this one "sideways".

Equation $\displaystyle {\color{blue}[1]}$ is: .$\displaystyle x \;=\;\frac{1}{2}y^2 - \frac{3}{2}$

Code:
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Substitute $\displaystyle {\color{blue}[2]}$ into $\displaystyle {\color{blue}[1]}$: .$\displaystyle y^2 \;=\;2y + 3\quad\Rightarrow\quad y^2 - 2y - 3 \;=\;0$

. . which factors: .$\displaystyle (y + 1)(y - 3) \;=\;0$

. . and has roots: .$\displaystyle y \:=\:\text{-}1,\,3$

The area is: .$\displaystyle A \;=\;\int_{-1}^3\left[y - \left(\frac{1}{2}y^2 - \frac{3}{2}\right)\right]\,dy$

Got it?

4. i got it

thanks... now the solution is complete