Find the area common to the parabolas y2 = 4x and x2 = 4y.
Find the area common to the parabolas y2=4xy^2=4xy2=4x and x2=4yx^2=4yx2=4y .
Solution:
Points of interception:
{y=±2xy=14x2\begin{cases} y=\pm2\sqrt x \\ y=\frac14x^2 \end{cases}{y=±2xy=41x2
{x=0y=0\begin{cases} x=0 \\ y=0 \end{cases}{x=0y=0 and {x=4y=4\begin{cases} x=4 \\ y=4 \end{cases}{x=4y=4 .
Area:
S=∫04(2x−14x2)dx=S=\displaystyle\int_0^4 (2\sqrt x-\frac14x^2)dx=S=∫04(2x−41x2)dx= (2⋅23⋅x32−14⋅13⋅x3)∣04=253−423=163(2\cdot \frac23\cdot x^\frac32-\frac14\cdot\frac13\cdot x^3)|_0^4=\frac{2^5}{3}-\frac{4^2}{3}=\frac{16}{3}(2⋅32⋅x23−41⋅31⋅x3)∣04=325−342=316 .
Answer: S=163S=\frac{16}{3}S=316 .
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