2 What If … ? We want to find the area between f(x) and g(x) ? Any ideas?
3 When f(x) < 0Consider taking the definite integral for the function shown below.The integral gives a negative area (!?)We need to think of this in a different wayabf(x)
4 Recall our look at odd functions on the interval [-a, a] Another ProblemWhat about the area between the curve and the x-axis for y = x3What do you get for the integral?Since this makes no sense – we need another way to look at itRecall our look at odd functions on the interval [-a, a]
5 Solution We can use one of the properties of integrals We will integrate separately for -2 < x < 0 and 0 < x < 2We take the absolute value for the interval which would give us a negative area.
6 General SolutionWhen determining the area between a function and the x-axisGraph the function firstNote the zeros of the functionSplit the function into portions where f(x) > 0 and f(x) < 0Where f(x) < 0, take absolute value of the definite integral
7 Try This!Find the area between the function h(x)=x2 + x – 6 and the x-axisNote that we are not given the limits of integrationWe must determine zeros to find limitsAlso must take absolute value of the integral since specified interval has f(x) < 0
8 Area Between Two Curves Consider the region between f(x) = x2 – 4 and g(x) = 8 – 2x2Must graph to determine limitsNow consider function inside integralHeight of a slice is g(x) – f(x)So the integral is
9 The Area of a Shark Fin Consider the region enclosed by Again, we must split the region into two parts0 < x < 1 and 1 < x < 9
10 Slicing the Shark the Other Way We could make these graphs as functions of yNow each slice is y by (k(y) – j(y))
11 Practice Determine the region bounded between the given curves Find the area of the region
12 Horizontal SlicesGiven these two equations, determine the area of the region bounded by the two curvesNote they are x in terms of y
14 Integration as an Accumulation Process Consider the area under the curve y = sin xThink of integrating as an accumulation of the areas of the rectangles from 0 to bb
15 Integration as an Accumulation Process We can think of this as a function of bThis gives us the accumulated area under the curve on the interval [0, b]
16 Try It OutFind the accumulation function forEvaluateF(0)F(4)F(6)
17 ApplicationsThe surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +kDetermine the value for k if the two functions are tangent to one anotherFind the area of the surface of the machine part