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We can derive a formula for the surface area much as we derived the formula for arc length. We’ll start by dividing the interval into \(n\) equal subintervals of width \(\Delta x\). On each subinterval we will approximate the function with a straight line that agrees with the function at the endpoints of each interval.
In this section we will show how a double integral can be used to determine the surface area of the portion of a surface that is over a region in two dimensional space.
Learning Objectives. 6.4.1 Determine the length of a curve, y = f ( x ) , between two points. 6.4.2 Determine the length of a curve, x = g ( y ) , between two points. 6.4.3 Find the surface area of a solid of revolution. In this section, we use definite integrals to find the arc length of a curve.
Let’s now use this formula to calculate the surface area of each of the bands formed by revolving the line segments around the \(x\)-axis. A representative band is shown in the following figure. Figure \(\PageIndex{9}\): A representative band used for determining surface area.
Let’s now use this formula to calculate the surface area of each of the bands formed by revolving the line segments around the \(x\)-axis. A representative band is shown in the following figure. Figure \(\PageIndex{9}\): A representative band used for determining surface area.
Determine the length of a curve, [latex]y=f (x), [/latex] between two points. Determine the length of a curve, [latex]x=g (y), [/latex] between two points. Find the surface area of a solid of revolution. In this section, we use definite integrals to find the arc length of a curve.
Example \(\PageIndex{3}\): Finding the surface area of a cone. The general formula for a right cone with height \(h\) and base radius \(a\) is \( f(x,y) = h-\dfrac{h}a\sqrt{x^2+y^2},\) shown in Figure 13.34. Find the surface area of this cone. Figure \(\PageIndex{3}\): Finding the surface area of a cone in Example \(\PageIndex{3}\). Solution
In this section we will discuss how to find the surface area of a solid obtained by rotating a parametric curve about the x or y-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation).
Introduction to Surface Area. proportional to the length ds of the segment. In our example, the total surface area swept out by a small segment of arc will be: dA =. (2πy) (ds). . circumference.
Surface Area of a Surface of Revolution. Let f(x) be a nonnegative smooth function over the interval [a, b]. Then, the surface area of the surface of revolution formed by revolving the graph of f(x) around the x -axis is given by. Surface Area = ∫b a(2πf(x)√1 + (f(x))2)dx.