Intereting Posts

An $r\times r$ submatrix of independent rows and independent columns is invertible (Michael Artin's book).
Find the parametric form $S(u, v)$ where $a \le u \le b$ and $c \le v \le d$ for the triangle with vertices $(1, 1, 1), (4, 2, 1),$ and $(1, 2, 2)$.
Showing a recursion sequence isn't bounded $a_{n+1}=a_n+\frac 1 {a_n}$
Cardinality of Borel sigma algebra
What's the name for the property of a function $f$ that means $f(f(x))=x$?
How to find $\lim\limits_{n\to\infty} n·(\sqrt{a}-1)$?
Do the two limits coincide?
Proving “The sum of n consecutive cubes is equal to the square of the sum of the first n numbers.”
How to Prove the Root Test For Series
Integral $\int_0^\infty\left(x+5\,x^5\right)\operatorname{erfc}\left(x+x^5\right)\,dx$
How to find the subgroups of S4 generated by these sets.
Weak categoricity in first order logic
Group Theory – Proving $(a*b)^{-1} = (a^{-1}) * (b^{-1}) $
Evaluate $\int\frac{1}{1+x^6} \,dx$
Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without using calculus, with directly using geometry. I think it would be very interesting challenge, but I am not able to think of a way out.

- Area and Polar Coordinates
- Triangles packed into a unit circle
- How to calculate the area of a region with a closed plane curve boundary?
- How to maximize area of two circles inside a rectangle without overlapping?
- How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$
- find maximum area

- general equation of a tangent line to a hyperbola
- Circle areas on squared grid
- Area of the field that the cow can graze.
- Plot $|z - i| + |z + i| = 16$ on the complex plane
- How do you find the distance between two points on a parabola
- How to convert the general form of ellipse equation to the standard form?
- How to get the limits of rotated ellipse?
- How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$
- How to find an ellipse , given 2 passing points and the tangents at them?
- How to geometrically prove the focal property of ellipse?

It depends a little bit what you mean by geometry. If you can see “geometrically” that stretching the function horizontally by a factor of $2$ should double the value of the integral, and if you can see “geometrically” that the integral of a sum of two functions should be the sum of the integrals, then there is such a proof, and I spell it out here:

https://mathoverflow.net/questions/114738/integrating-powers-without-much-calculus/114843#114843

Using a similar transformation used by Archimedes for sphere and cylinder, show equivalence of slice of the curve at point $x$ to the area of a pyramid slice. The total area will be equal to the volume of the pyramid of unit base and height. This is however equivalent to calculus (under disguise.)

- Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category
- Area of intersection between two circles
- Representation of symmetric functions
- Construct a finite field of order 27
- The closure of a product is the product of closures?
- Spectrum of idempotent element
- Subspace of $C^3$ that spanned by a set over C and over R
- $AB$ is not invertible
- How to differentiate Complex Fluid Potential
- The Group of order $p^3$
- Derivative of a Matrix with respect to a vector
- Why is a circle 1-dimensional?
- In what senses are archimedean places infinite?
- Eigenvalues of an $n\times n$ symmetric matrix
- What is the largest $k$ such that $ \frac { k(abc) }{ a+b+c } \le \left( a+b \right) ^{ 2 }+\left( a+b+4c \right) ^{ 2 } $?