User:DemonThing/MathScratchpad

This page is a scratchpad such as, but for assisting me in helping people with math topics.

(I hope this kind of thing is allowed.) =1= $$y^2 = -4x$$

$$x = f(y) = - \frac{1}{4} y^2$$

$$y^2 = -2(x-1)$$

$$x = g(y) = 1 - \frac{1}{2} y^2$$

$$A = \int_{-2}^{2} g(y) - f(y) \, dy$$

$$A = \int_{-2}^{2} (1 - \frac{1}{2} y^2) - (- \frac{1}{4} y^2) \, dy$$

$$A = \int_{-2}^{2} 1 - \frac{1}{2} y^2 + \frac{1}{4} y^2 \, dy$$

$$A = \int_{-2}^{2} 1 - \frac{1}{4} y^2 \, dy$$

$$A = \int_{-2}^{2} dy - \int_{-2}^{2} \frac{1}{4} y^2 \, dy$$

$$A = \int_{-2}^{2} dy - \frac{1}{4} \int_{-2}^{2} y^2 \, dy$$

$$A = \left[ y - \frac{1}{4} \left( \frac{1}{3} y^3 \right) \right]_{-2}^{2}$$

$$A = \left( 2 - \frac{1}{12} 2^3 \right) - \left( -2 - \frac{1}{12} (-2)^3 \right)$$

$$A = \left( 2 - \frac{2}{3} \right) - \left( -2 - \frac{-2}{3} \right )$$

$$A = \frac{4}{3} - \left( -\frac{4}{3} \right)$$

$$A = \frac{8}{3}$$

=2= $$f(x) = g(x)$$ where $$f(x) = |x| - x$$ and $$g(x) = -x^2 - x + 2$$