Area using $\iint dxdy$

Yes, consider $f(x,y)=1$. The volume under this curve is $1$ (the height) times the area of the base, so the numerical value of volume coincides with the area: they're the same thing.


If $B$ is a measurable and bounded subset of $ \mathbb R^2$, then the area of $B$ is given by

$\iint_B 1 d(x,y).$


One of the most common ways to think of a double integral is to think of it as the volume under a surface created by a function of two variables $z=f(x,y)$ and bounded by a region $R$ (typically rectangular in shape) in the $xy$ plane. If the given function of two variables is $f(x,y)=1$, then what you're basically doing is you're finding the volume of a solid whose height is $1$ and the base area is the area of the given region $R$. The volume of a solid whose height is $1$ is numerically equal to its base area. That's why

$$\iint_{R}1\,dx\,dy=area\ of\ region\ R.$$

Tags:

Integration

Multivariable Calculus

Related

In a right angled $\triangle ABC$, $DE$ and $DF$ are perpendicular to $AB$ and $BC$ respectively. What is the probability of $DE\cdot DF>3$? Why these $2$ methods give the exact same answer for sum of squares? Computing sum of lengths of legs of a right triangle Show that for any prime $p$ and any integer $m$, $m^p + (p − 1)! m$ is divisible by $p$. Evaluating $\frac{2013^3-2\cdot 2013^2\cdot 2014+3\cdot 2013\cdot 2014^2-2014^3+1}{2013\cdot 2014}$ Show that $\int_{-\infty}^{\infty} \frac{\cos (\theta) e^{\theta y}}{2 \cosh(\pi y/2)} y dy=\tan (\theta)$. I need help computing $\int {\ln x\over 2-x}\, dx$ If $L\mid K$ is a finite extension of fields then K is perfect iff L is perfect Applications of Ramanujan's Master Theorem Perimeter of orthic triangle List of small rings Is a dense set always infinite?

Recent Posts