Hello everyone,
I want to calculate the area of a radius (r = 15) by integrating.
How do I get this? Who can help me get started?
Thanks.
Roy
I assume you mean the surface area of a sphere, $S$ ...
$\displaystyle S = 2\pi \int_{-r}^r y \cdot \sqrt{1+\left(\dfrac{dy}{dx}\right)^2} \, dx$
where $y= \sqrt{r^2-x^2}$
... note that there is an opportunity to take advantage of the sphere's symmetry by doubling the value of the definite integral from $x=0$ to $x=r$.
This is just a the area between a square of side length 15 and one fourth of a circle of radius 15 ...
$A = 15^2 - \dfrac{\pi \cdot 15^2}{4}$
Your sketch is obviously not to scale. Why do you require integration to find this?
Oh, well ...
$\displaystyle A= \int_0^{15} 15-\sqrt{15^2-x^2} \, dx$
... and as chiro stated, a trig substitution can be used for integrating the radical expression.
Let $x=15\sin{t}$
Thanks for your comment.
Well, I need to determine the surface of an angle section ( 200/100/12 ) , but the formula for this must apply to each angle section.
I am busy with the chapter integrating. So I assumed we had to calculate this area by means of an integral.
How does this work with a trig substitution ?
$\displaystyle A = \int_0^{15} 15 - \sqrt{15^2 - x^2} \, dx = \int_0^{15} 15 \, dx - \int_0^{15} \sqrt{15^2-x^2} \, dx = 15^2 - \int_0^{15} \sqrt{15^2-x^2} \, dx$
$x = 15\sin{t}$
$x=0 \implies t = 0$
$x=15 \implies t = \dfrac{\pi}{2}$
$dx = 15\cos{t} \, dt$
substitute ...
$\displaystyle A = 15^2 - \int_0^{\pi/2} \sqrt{15^2 - 15^2\sin^2{t}} \cdot 15\cos{t}\, dt$
$\displaystyle A = 15^2 - 15^2\int_0^{\pi/2} \sqrt{1 - \sin^2{t}} \cdot \cos{t}\, dt$
$\displaystyle A = 15^2 - 15^2\int_0^{\pi/2} \cos^2{t}\, dt$
$\displaystyle A = 15^2 - \dfrac{15^2}{2}\int_0^{\pi/2} 1 + \cos(2t) \, dt$
$A = 15^2 - \dfrac{15^2}{2} \bigg[t + \dfrac{\sin(2t)}{2} \bigg]_0^{\pi/2}$
$A = 15^2 - \dfrac{15^2}{2} \bigg[\dfrac{\pi}{2} - 0 \bigg]$
$A = 15^2 - \dfrac{15^2 \pi}{4}$
I need to determine a formula that we can determine the red shaded area. ( see picture below )
a=200 , b=100 and t=12. ( But these values could also be different )
I thought, if I first determine the integral of the radius, I'll figure it out. But how do I process this in formula for the entire area?
I suggest: A= (a*t-r2) + ((b-t)*t-r2) . But how do I make an integral formula of A?
Are you required to do this using an integral? This reduces, as before, to rectangles and circles and you can calculate those areas from the usual geometric formulas. I see this as two quarter circles, with radius , at each end. That has area . There is a rectangle, on the left, with length and width t. That has area . There is a rectangle, at the bottom, with length and width t. That has area .