MAT01B1: Surface Area of Solids of Revolution

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1 MAT01B1: Surface Area of Solids of Revolution Dr Craig 02 October 2018

2 My details: Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring (Or, just google Andrew Craig maths.)

3 Collect tut assessments Class Test 4 and Assignment 3 are available for collection.

4 From last week: Arc length of a curve y = f(x), for a x b, can be calculated using the following formula: L = = b a b a 1 + ( ) 2 dy dx dx 1 + [f (x)] 2 dx

5 From last week: Arc length of a curve x = g(y), for c y d can be calculated using the following formula: L = = d c d c 1 + ( ) 2 dx dy dy 1 + [g (y)] 2 dy

6 Arc length function The following equation calculates the length of the arc of a smooth curve f(x) from the initial point (a, f(a)) to the point (x, f(x)) (where a x b). Now s(x) = x ds dx = 1 + [f (x)] 2 = a 1 + [f (t)] 2 dt 1 + ( ) 2 dy dx

7 Example: find the arc length function for the curve y = x ln x taking (1, 1) as the starting point.

8 Surface area

9 Using arc length to find surface area: We will now look at the surface area of the solid that is obtained when a curve is rotated around the x- or y-axis. To do this, we must integrate along the curve.

10 Introduction to Surface Area The approximating surface consists of a number of bands, each formed by rotating a line segment about an axis.

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12 Surface area of two familiar solids: We are interested in the lateral surface area. This means the surface area of a solid of revolution without including the surfaces at the ends of the solid. It is useful to recall the formulas for the lateral surface area of the following shapes: Cylinder: A=circumference height A = 2πrh Cone: A = πr r 2 + h 2

13 Suppose that f(x) is a positive function and has a continuous derivative. The lateral surface area of the surface obtained by rotating the curve y = f(x), a x b about the x-axis is S = b a 2πf(x) 1 + [f (x)] 2 dx If we have x = g(y), c y d then d ( ) 2 dx S = 2πy 1 + dy dy c

14 Example: Find the surface area of the line y = 3x with 0 x 3 rotated about the x-axis. Do this in two different ways and then check it using the formula for the surface area of a cone.

15 Example: the curve y = 4 x 2, 1 x 1 is an arc of the circle x 2 + y 2 = 4. Find the area when this arc is rotated about the x-axis.

16 Suppose that f(x) is a positive function and has a continuous derivative. The surface area of the surface obtained by rotating the curve y = f(x), a x b about the y-axis is S = b a 2πx 1 + [f (x)] 2 dx If we have x = g(y), c y d then d ( ) 2 dx S = 2πg(y) 1 + dy dy c

17 Example: Rotate the arc of the parabola y = x 2 from (1, 1) to (2, 4) about the y-axis. Find the resulting surface area. Note: there are two possible methods for solving this problem. Solution: π 6 ( ) 5.

18 Example: Find the surface area when the portion of the curve y = 3 3x when 0 y 2 is rotated about the y-axis.

19 Example: Find the surface area of the solid formed by rotating y = 2x + 1 (0 x 2) around the x-axis.

20 Example: Find the area of the surface generated by rotating the curve y = e x, 0 x 1, about the x-axis.

21 The previous example can be done using either arc length calculation. If you do it by integrating with respect to x, you will probably use u-substitution to help with the integration. The integrand after this substitution is the same integrand (but in terms of u) that you get if you integrate with respect to y.

22 Stretch yourself: Do Exercise 63 from Ch 7.8 and then Exercise 27 from Ch 8.2.