Application of Double Integration

Applications of Double Integrals

Area Enclosed by Plane Curves

Double integration is used to compute area, volume, mass and many physical quantities.

$$A = \iint_R 1\, dA$$

Example 1

Find the area of a quadrant of $x^2 + y^2 = a^2$.

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Using vertical strips: $$A=\int_0^a \int_0^{\sqrt{a^2-x^2}} dy\,dx$$

$$A=\int_0^a \sqrt{a^2-x^2}\,dx$$

Using standard integral: $$\int \sqrt{a^2-x^2}dx =\frac{x}{2}\sqrt{a^2-x^2} +\frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right)$$

Evaluating limits: $$A=\frac{\pi a^2}{4}$$

Final Answer: $$\boxed{\frac{\pi a^2}{4}}$$

Example 2

Find the area bounded by $y=2-x$, $y^2=2(x+2)$.

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Intersection points: $$(2-x)^2=2(x+2)$$ $$x=0,6$$

Using horizontal strips: $$A=\int_{-4}^{2}\int_{(y^2/2)-2}^{2-y} dx\,dy$$

$$A=\int_{-4}^{2}\left(4-y-\frac{y^2}{2}\right) dy$$

After evaluation: $$A=18$$

Final Answer: $$\boxed{18}$$

Example 3

Find the area bounded by $2x-3y+4=0$, $x+y-3=0$, $y=0$.

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Vertices: $$(-2,0), (3,0), (1,2)$$

$$A=\int_{0}^{2}\int_{\frac{3y-4}{2}}^{3-y} dx\,dy$$

$$A=\int_{0}^{2}\left(5-\frac{5y}{2}\right) dy$$

$$A=5$$

Final Answer: $$\boxed{5}$$

Example 4

Find the area bounded by $x^2=4(y+2)$ and $x^2=3-y$.

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Intersection: $$4(y+2)=3-y$$ $$y=-1,\quad x=\pm2$$

$$A=\int_{-2}^{2}\int_{\frac{x^2}{4}-2}^{3-x^2} dy\,dx$$

$$A=\int_{-2}^{2}\left(5-\frac{5x^2}{4}\right) dx$$

$$A=\frac{172}{15}$$

Final Answer: $$\boxed{\frac{172}{15}}$$

Example 5

Find the area bounded by $x(x^2+y^2)=a(x^2-y^2)$.

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Using polar substitution: $$r = a\cos(2\theta)\sec\theta$$

Area of full loop: $$A=4a^2\int_{0}^{\pi/2} \left(\sin^2\frac{\theta}{2} -2\sin^4\frac{\theta}{2}\right)d\theta$$

After evaluation: $$A=\frac{\pi a^2}{2}$$

Final Answer: $$\boxed{\frac{\pi a^2}{2}}$$

Conclusion

Double integrals are powerful tools for computing area, volume, and many physical applications.