Find the area of the region enclosed between the line y=x and the parabola y=x².

Worked Examples · Example 6

y=x

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Step 1 — find the points of intersection. Set the two equations equal:

x=x^{2}\ \Rightarrow\ x^{2}-x=0\ \Rightarrow\ x(x-1)=0\ \Rightarrow\ x=0\ \text{or}\ x=1

So the limits of integration are $a=0$ and $b=1$ .

Step 2 — decide which curve is on top. Test $x=\tfrac{1}{2}$ : the line gives $y=\tfrac{1}{2}=0.5$ , the parabola gives $y=\left(\tfrac{1}{2}\right)^{2}=0.25$ . The line $y=x$ is above the parabola $y=x^{2}$ on $(0,1)$ , so upper $=x$ , lower $=x^{2}$ .

Step 3 — integrate the difference (upper $-$ lower). …

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Q.Find the area of the region enclosed between the line $y=x$ and the parabola $y=x^{2}$ .

Concept understanding — Area between Two Curves

Q.Find the area of the region enclosed between the line y=xy=xy=x and the parabola y=x2y=x^{2}y=x2.

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Q.Find the area of the region enclosed between the line $y=x$ and the parabola $y=x^{2}$ .

Unlock everything free for 14 days