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.

Problem of the Day - CalculatorLet f be the function given by f(x) = 3e3x

and let g be the function given by g(x) = 6x3. At what value of x do the graphs of f and g have parallel tangent lines?

A) -0.701B) -0.567C) -0.391D) -0.302E) -0.258

Problem of the DayLet f be the function given by f(x) = 3e3x

and let g be the function given by g(x) = 6x3. At what value of x do the graphs of f and g have parallel tangent lines?

A) -0.701B) -0.567C) -0.391D) -0.302E) -0.258

Take derivatives and set equal to each other. Find the zero of that function.

Area thus far -

x

y g(x)Area thus far -

Area =

Theorem 6.1The 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]

Example for non-intersection curves - Find the area between y = sec2x and y = sin x from x = 0 to π/41. Graph to see which is above

2. Integrate

sec2x

sin x

Example for intersecting curves - Find the area enclosed by y = 2 - x2 and y = -x 1. Graph to see which is above

(2 - x2)

2. Find points of intersection

y = y2 - x2 = -x - 20 = x2 - x - 20 = (x - 2)(x + 1) x = 2 or -1

Example for intersecting curves - Find the area enclosed by y = 2 - x2 and y = -x

3. Integrate

2. Find points of intersection

x = 2 or -1

+

Example for curves that intersect at more than 2 points Find the area enclosed by f(x) = 3x3 - x2 - 10x and g(x) = -x2 + 2x1. Graph to see which is above f(x) = 3x3 - x2 -

10x

g(x) = -x2 + 2x

Example for curves that intersect at more than 2 points Find the area enclosed by f(x) = 3x3 - x2 - 10x and g(x) = -x2 + 2x2. Find points of intersection f(x) = 3x3 - x2 -

10x

g(x) = -x2 + 2x

y = y3x3 - x2 - 10x = -x2 + 2x3x3 - 12x = 03x(x2 - 4) = 03x(x - 2)(x + 2) = 0x = 0, -2, 2

3. Integrate

f(x) = 3x3 - x2 - 10x

g(x) = -x2 + 2x-2

0

2

From -2 to 0 f(x) is on topFrom 0 to 2 g(x) is on top

3. Integrate

f(x) = 3x3 - x2 - 10x

g(x) = -x2 + 2x

Horizontal rectangles and integration with respect to yThus far -

(vertical rectangles in x)

(horizontal rectangles in y)

To do horizontal -

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

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 =

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

Example - Find the area of the region in Quadrant I bounded above by y = , below by the x-axis, and y = x - 23. Integrate

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