1. P roblem of the Day - Calculator Let f be the function given by f(x) = 3e 3x and let g be the function given by g(x) = 6x 3. At what value of x do the graphs of f and g have parallel tangent lines? A) -0.701 B) -0.567 C) -0.391 D) -0.302 E) -0.258

2. P roblem of the Day Let f be the function given by f(x) = 3e 3x and let g be the function given by g(x) = 6x 3. At what value of x do the graphs of f and g have parallel tangent lines? A) -0.701 B) -0.567 C) -0.391 D) -0.302 E) -0.258 Take derivatives and set equal to each other. Find the zero of that function.

3. Area thus far -

4. x y g(x) Area thus far -

5. Area =

6. Theorem 6.1 The area of the region between two curves f(x) and g(x) bounded by x = a and x = b is if f and g are continuous on [a, b] and g(x) < f(x) in [a, b]

7. E xample for non-intersection curves - Find the area between y = sec 2 x and y = sin x from x = 0 to π/4 1. Graph to see which is above 2. Integrate sec 2 x sin x

8. E xample for intersecting curves - Find the area enclosed by y = 2 - x 2 and y = - x 1. Graph to see which is above (2 - x 2 ) 2. Find points of intersection y = y 2 - x 2 = -x - 2 0 = x 2 - x - 2 0 = (x - 2)(x + 1) x = 2 or -1

9. E xample for intersecting curves - Find the area enclosed by y = 2 - x 2 and y = - x 3. Integrate 2. Find points of intersection x = 2 or -1 +

10. E xample for curves that intersect at more than 2 points Find the area enclosed by f(x) = 3x 3 - x 2 - 10x and g(x) = -x 2 + 2x 1. Graph to see which is above f(x) = 3x 3 - x 2 - 10x g(x) = -x 2 + 2x

11. E xample for curves that intersect at more than 2 points Find the area enclosed by f(x) = 3x 3 - x 2 - 10x and g(x) = -x 2 + 2x 2. Find points of intersection f(x) = 3x 3 - x 2 - 10x g(x) = -x 2 + 2x y = y 3x 3 - x 2 - 10x = -x 2 + 2x 3x 3 - 12x = 0 3x(x 2 - 4) = 0 3x(x - 2)(x + 2) = 0 x = 0, -2, 2

12. 3. Integrate f(x) = 3x 3 - x 2 - 10x g(x) = -x 2 + 2x -2 0 2 From -2 to 0 f(x) is on top From 0 to 2 g(x) is on top

13. 3. Integrate f(x) = 3x 3 - x 2 - 10x g(x) = -x 2 + 2x

14. Horizontal rectangles and integration with respect to y Thus far - (vertical rectangles in x) (horizontal rectangles in y) To do horizontal -

15. Example - Find the area of the region in Quadrant I bounded above by y =, below by the x-axis, and y = x - 2 vertical rectangles would require more than 1 integration horizontal rectangles would require 1 integration

16. Example - Find the area of the region in Quadrant I bounded above by y =, below by the x-axis, and y = x - 2 1. Change equations to x =

17. Example - Find the area of the region in Quadrant I bounded above by y =, below by the x-axis, and y = x - 2 2. Find intersection

18. Example - Find the area of the region in Quadrant I bounded above by y =, below by the x-axis, and y = x - 2 3. Integrate

19. Attachments