The Area Between Two Curves Lesson 6.1

What If … ? We want to find the area between f(x) and g(x) ? Any ideas?

When f(x) < 0 Consider taking the definite integral for the function shown below. The integral gives a negative area (!?) We need to think of this in a different way a b f(x)

Recall our look at odd functions on the interval [-a, a] Another Problem What about the area between the curve and the x-axis for y = x3 What do you get for the integral? Since this makes no sense – we need another way to look at it Recall our look at odd functions on the interval [-a, a]

Solution We can use one of the properties of integrals We will integrate separately for -2 < x < 0 and 0 < x < 2 We take the absolute value for the interval which would give us a negative area.

General Solution When determining the area between a function and the x-axis Graph the function first Note the zeros of the function Split the function into portions where f(x) > 0 and f(x) < 0 Where f(x) < 0, take absolute value of the definite integral

Try This! Find the area between the function h(x)=x2 + x – 6 and the x-axis Note that we are not given the limits of integration We must determine zeros to find limits Also must take absolute value of the integral since specified interval has f(x) < 0

Area Between Two Curves Consider the region between f(x) = x2 – 4 and g(x) = 8 – 2x2 Must graph to determine limits Now consider function inside integral Height of a slice is g(x) – f(x) So the integral is

The Area of a Shark Fin Consider the region enclosed by Again, we must split the region into two parts 0 < x < 1 and 1 < x < 9

Slicing the Shark the Other Way We could make these graphs as functions of y Now each slice is y by (k(y) – j(y))

Practice Determine the region bounded between the given curves Find the area of the region

Horizontal Slices Given these two equations, determine the area of the region bounded by the two curves Note they are x in terms of y

Assignments A Lesson 7.1A Page 452 Exercises 1 – 45 EOO

Integration as an Accumulation Process Consider the area under the curve y = sin x Think of integrating as an accumulation of the areas of the rectangles from 0 to b b

Integration as an Accumulation Process We can think of this as a function of b This gives us the accumulated area under the curve on the interval [0, b] View nSpire Demo

Try It Out Find the accumulation function for Evaluate F(0) F(4) F(6)

Applications The surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +k Determine the value for k if the two functions are tangent to one another Find the area of the surface of the machine part

Assignments B Lesson 7.1B Page 453 Exercises 57 – 65 odd, 85, 88