Area between triangle and parabola

Question

Consider a triangle of vertices $(-3,2),(1,4),(3,1)$

Describe the area of region between the $y=x^2$ and inside of the triangle mentioned above as a definite integral.

Can someone help me solve this problem.

Answer 1

Well, if you plot the given data we get the following figure:

When we connect the data points $(-3,2)$ and $(3,1)$ with a line, we get the following equation:

$$ \begin{cases} 2=\text{a}\cdot\left(-3\right)+\text{b}\\ \\ 1=\text{a}\cdot3+\text{b} \end{cases}\space\space\space\Longleftrightarrow\space\space\space\begin{cases} \text{a}=-\frac{1}{6}\\ \\ \text{b}=\frac{3}{2} \end{cases}\space\space\space\therefore\space\space\space\text{y}\left(x\right)=\frac{3}{2}-\frac{x}{6}\tag1 $$

Plotting this again gives the following figure:

Solving for the intersection points we get:

$$x^2=\frac{3}{2}-\frac{x}{6}\space\Longleftrightarrow\space x=\frac{\pm\sqrt{217}-1}{12}\tag2$$

Plotting these lines as verticals give the following figure:

Connecting these points with the last point of the triangle gives:

• $$ \begin{cases} 4=\text{a}\cdot1+\text{b}\\ \\ \left(\frac{\sqrt{217}-1}{12}\right)^2=\text{a}\cdot\frac{\sqrt{217}-1}{12}+\text{b} \end{cases}\space\space\space\Longleftrightarrow\space\space\space\begin{cases} \text{a}=-\frac{4\sqrt{217}+53}{6}\\ \\ \text{b}=\frac{4 \sqrt{217}+77}{6} \end{cases}\space\space\space\therefore\space\space\space$$ $$\text{m}\left(x\right)=\frac{4 \sqrt{217}+77}{6}-\frac{4\sqrt{217}+53}{6}\cdot x\tag3$$
• $$ \begin{cases} 4=\text{a}\cdot1+\text{b}\\ \\ \left(\frac{-\sqrt{217}-1}{12}\right)^2=\text{a}\cdot\left(\frac{-\sqrt{217}-1}{12}\right)+\text{b} \end{cases}\space\space\space\Longleftrightarrow\space\space\space\begin{cases} \text{a}=\frac{4 \sqrt{217}-53}{6}\\ \\ \text{b}=\frac{77-4 \sqrt{217}}{6} \end{cases}\space\space\space\therefore\space\space\space$$ $$\text{n}\left(x\right)=\frac{4 \sqrt{217}-53}{6}\cdot x+\frac{77-4 \sqrt{217}}{6}\tag4$$

Plotting these lines gives:

Now, I let you conclude.

Answer 2

Find the pts of intersection of triangle and parabola and the find the area under line using application of integration