Find the area of the region bounded by the parabola $x^2 = y$ and the line $y = x + 2$.

Find the area of the region bounded by the parabola x² = y and the line y = x + 2.

Exercise: Area Between Two Curves · Q21

x^2 = y

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The curves meet at $x=-1$ and $x=2$ ; the line lies above the parabola between them.

Setting $x^2=x+2$ gives $x^2-x-2=0 \Rightarrow (x-2)(x+1)=0$ , so $x=-1$ or $x=2$ . Testing $x=0$ : the line gives $y=2$ and the parabola gives $y=0$ , so the line $y=x+2$ lies above the parabola $y=x^2$ throughout $(-1,2)$ . The area is

\text{Area} = \int_{-1}^{2}\big[(x+2)-x^2\big]\,dx = \left[\frac{x^2}{2}+2x-\frac{x^3}{3}\right]_{-1}^{2}.

At $x=2$ : $\dfrac{4}{2}+4-\dfrac{8}{3} = 2+4-\dfrac{8}{3} = 6-\dfrac{8}{3}=\dfrac{10}{3}$ . …

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Q.Find the area of the region bounded by the parabola $x^2 = y$ and the line $y = x + 2$ .

Concept understanding — Area between Two Curves

Q.Find the area of the region bounded by the parabola x2=yx^2 = yx2=y and the line y=x+2y = x + 2y=x+2.

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Q.Find the area of the region bounded by the parabola $x^2 = y$ and the line $y = x + 2$ .

Unlock everything free for 14 days