APPLICATI

ONS OF

INTEGRATION

A R E A BE T W E E N 2

CU R V E S

AREA BETWEEN 2 CURVESWe want to find the area between f(x)

= x2 and g(x) = -x + 6.

Always graph the two functions first to get a visual of the area.

FIND POINTS OF INTERSECTIONBecause we only want the area between,

we have to find the points of intersection.

To do this, set f(x) = g(x) and solve for x.x2 = -x + 6

x = -3 and 2

FIND THE AREA UNDER BOTH CURVESThe area under f(x) = .The area under g(x) = .

If we subtract the area under f(x) from the area under g(x), we are left with the area between the two curves.

FORMULAIf f(x) > g(x) on [a, b], then the area

between f(x) and g(x) is

left point of intersection

right point of intersection

upper function

lower function

PRACTICESet up and solve the integral to find

the area between f(x) = 2 – x2 and g(x) = -x.

Area is 9/2 units squared

TO USE THE CALCULATOR

Input the two equations into y1 and y2.Graph and find the two intersections.

2nd CALC intersect ENTER “First curve?” ENTER “Second curve?” ENTER “Guess?” Move cursor over one intersection point, ENTER Get solution, repeat steps for second intersection point

x = -1 and 2

TO USE THE CALCULATORBecause you have seen the graph, you know

that the parabola is the upper function and the line is the lower function. You also know a and b, therefore you just have to write the integral and input it into the calculator.

A = MATH fnInt(ENTERInput the lower and upper limits and the integrand into the

boxes and ENTER

Record the solution

YOUR TURN!Fill in the blanks for each problem.

1. Graph

2. Intersection points: x = _________3. Upper function: _______________ Lower function: _______________4. Integral: _____________________5. Area: ________________________

PROBLEMS1. Find the area of the region enclosed by

f(x) = 2 cos x and g(x) = x2 – 1A = 4.99 un2

2. Find the area of the region enclosed by f(x) = 7 – 2x2 and g(x) = x2 + 4

A = 4 un2

ASSIGNMENTpage 452 1 – 6, 20 – 55 by 5