Length and Surface Area of Plane Curves: Definition, Exercises, and Formulas, Summaries of Analytical Geometry and Calculus

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Length of a Curve and

Surface Area

Length of a Plane Curve A plane curve is a curve that lies in a two-dimensional plane. We can define a plane curve using parametric equations. This means we define both x and y as functions of a parameter. Parametric equations Definition A plane curve is smooth if it is given by a pair of parametric equations x =f(t) , and y =g(t) , t is on the interval [ a,b ] where f' and g' exist and are continuous on [ a,b ] and f'(t) and g'(t) are not simultaneously zero on ( a,b ).

Arc length We can approximate the length of a plane curve by adding up lengths of linear segments between points on the curve.

• EX 2 Find the circumference of the circle x^2 + y^2 = r
• EX 4 Find the arc length of the curve f(x) = √x from x = 0 to x =

Surface Area Differential of Arc Length Let f(x) be continuously differentiable on [a,b]. Start measuring arc length from (a,f(a)) up to (x,f(x)), where a is a real number. Then, the arc length is a function of x.

EX 4 Find the area of the surface generated by revolving y = √ 25 - x^2 on the interval [ - 2 , 3 ] about the x-axis.

EX 5 Find the area of the surface generated by revolving x = 1 - t^2 , y = 2 t , on the t-interval [ 0 , 1 ] about the x-axis.