Calculus: Area. Mathematics 15: Lecture 22. Dan Sloughter. Furman University. November 12, 2006

Transcription

1 Calculus: Area Mathematics 15: Lecture 22 Dan Sloughter Furman University November 12, 2006 Dan Sloughter (Furman University) Calculus: Area November 12, / 7

2 Area Note: formulas for the areas of figures like rectangles, triangles, and trapezoids are relatively easy to derive. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

3 Area Note: formulas for the areas of figures like rectangles, triangles, and trapezoids are relatively easy to derive. Since any polygon may be divided up into triangles, it is a straightforward matter to compute the area of any polygon. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

4 Area Note: formulas for the areas of figures like rectangles, triangles, and trapezoids are relatively easy to derive. Since any polygon may be divided up into triangles, it is a straightforward matter to compute the area of any polygon. Until the invention of the calculus, there was no general technique for finding the area of an arbitrary region in the plane. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

5 Area Note: formulas for the areas of figures like rectangles, triangles, and trapezoids are relatively easy to derive. Since any polygon may be divided up into triangles, it is a straightforward matter to compute the area of any polygon. Until the invention of the calculus, there was no general technique for finding the area of an arbitrary region in the plane. Greek mathematicians had found many formulas for particular types of regions (Archimedes even wrote a book on finding the areas of segments of parabolas), but their methods did not generalize. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

6 Area (cont d) Suppose y is a function of x and let C be the curve determined by this relationship. We will suppose the curve lies entirely above the horizontal axis. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

7 Area (cont d) Suppose y is a function of x and let C be the curve determined by this relationship. We will suppose the curve lies entirely above the horizontal axis. Let A be the area under the curve from some fixed point a up to x. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

8 Area (cont d) Suppose y is a function of x and let C be the curve determined by this relationship. We will suppose the curve lies entirely above the horizontal axis. Let A be the area under the curve from some fixed point a up to x. Key observation: if x increases from x to x + x, then the increase in A is approximately y x. That is, A y x. C y y x a x x + x Dan Sloughter (Furman University) Calculus: Area November 12, / 7

9 The fundamental theorem In fact, da dx = lim A x 0 x = y. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

10 The fundamental theorem In fact, da dx = lim A x 0 x = y. This is remarkable because it connects two concepts that seem at first unrelated, namely, instantaneous rate of change and area. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

11 The fundamental theorem In fact, da dx = lim A x 0 x = y. This is remarkable because it connects two concepts that seem at first unrelated, namely, instantaneous rate of change and area. Indeed, this result is known as the Fundamental Theorem of Calculus. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

12 Example Suppose we want to find the area under the parabola y = x 2 from x = 1 to x = Dan Sloughter (Furman University) Calculus: Area November 12, / 7

13 Example Suppose we want to find the area under the parabola y = x 2 from x = 1 to x = If we let A be the area under the curve from x = 1 to an arbitrary x, then we know that da dx = x 2. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

14 Example (cont d) From what we know about derivatives, it follows that A = 1 3 x 3 + c for some constant c. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

15 Example (cont d) From what we know about derivatives, it follows that A = 1 3 x 3 + c for some constant c. Now we know that A = 0 when x = 1, and so 0 = A = c = c. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

16 Example (cont d) From what we know about derivatives, it follows that A = 1 3 x 3 + c for some constant c. Now we know that A = 0 when x = 1, and so 0 = A = c = c. Hence c = 1 3, and A = 1 3 x Dan Sloughter (Furman University) Calculus: Area November 12, / 7

17 Example (cont d) From what we know about derivatives, it follows that A = 1 3 x 3 + c for some constant c. Now we know that A = 0 when x = 1, and so 0 = A = c = c. Hence c = 1 3, and A = 1 3 x Thus the area from x = 1 to x = 4 is A = = 63 3 = 21. Dan Sloughter (Furman University) Calculus: Area November 12, / 7

18 Problems 1. Let C be the parabola with equation y = x 2. a. Find the function A which gives the area beneath C from x = 0 to an arbitrary x. b. Use the preceding result to find the area beneath C from x = 0 to x = 1. c. Find the function A which gives the area beneath C from x = 2 to an arbitrary x. d. Use the preceding result to find the area beneath C from x = 2 to x = Let C be the parabola with equation y = 3x 2. a. Find the function A which gives the area beneath C from x = 0 to an arbitrary x. b. Use the preceding result to find the area beneath C from x = 0 to x = 2. c. Find the function A which gives the area beneath C from x = 1 to an arbitrary x. d. Use the preceding result to find the area beneath C from x = 1 to x = 3. Dan Sloughter (Furman University) Calculus: Area November 12, / 7