advertisement

Section 6.4: Arc Length Elementary geometry doesn’t help to find the length of a curve that wiggles like this. So, let’s break it into pieces we can work with. Arc Length: y s 2 x 2 y 2 x s x y 2 Idea : y 2 1 2 x x 2 2 y dy x dx y 1 x 2 dy Reason MVT f '( xk* ) dx b a 1 f ' x dx 2 x f (x) = sinx, 0 x 0 1 cos x dx 2 2 f (x) = x on [0,3] 3 1 2x dx 2 0 Or, f (y) = y on [0,9] 9 0 1 1 2 y 2 dy Rotate the graph of x 2 about the x-axis. Find surface area. 9 3 2 ( x ) 2 0 3 1 2 x dx 2 9 Rotate about y = 4 x2 3 -4 3 2 ( x 0 2 4) 1 2 x dx 2 15 Rotate about y = 15 9 x2 3 3 2 (15 x ) 2 0 1 2 x dx 2