![So far this chapter we have used a representative element – in this case a
representative rectangle - to adapt integration to the area between curves. In
future sections we will continue to do the same thing. We will come up with a
representative element, then using precalculus and geometry formulas, adapt
integration to find an accumulation of those elements.
[ ( ) ( )]
b
a
f x g x dx−∫
For example, in this section we used the idea that Area = (height)(width) and
we found the area of a representative rectangle to show that
∆Area = [f(x) - g(x)]Δx. Then we summed up all those representative
rectangles with integration to get](https://image.slidesharecdn.com/calc7-1b-120130214107-phpapp01/85/Calc-7-1b-2-320.jpg)
![Ex 6 p.451 Describing Integration as an Accumulation Process
Find the area of the region bounded by the graph of y = 4 – x2
and the x-
axis. Describe the integration as an accumulation process.
( )
2
2
2
[ 4 0]x dx
−
− −∫The parabola will intersect the x-axis at x = 2
( )
2
2
2
4 0x dx
−
−
− =∫
As we let the representative rectangle move from left
to right, we can see area accumulating.
( )
1
2
2
5
4
3
x dx
−
−
− =∫
( )
0
2
2
16
4
3
x dx
−
− =∫
( )
1
2
2
27
4
3
x dx
−
− =∫
( )
2
2
2
32
4
3
x dx
−
− =∫](https://image.slidesharecdn.com/calc7-1b-120130214107-phpapp01/85/Calc-7-1b-3-320.jpg)
![Another example. Find area when horizontal rectangle makes sense.
( ) (2 )f y y y= − ( )g y y= −
Graph. Find intersections by setting f(y)=g(y). 2y – y2
= -y 3y – y2
= 0
y = 0, y = 3
(2 )x y y= − 2
2y y= − +
2
( 2 1 1)y y= − − + −
2
(( 1) 1)y= − − −
2
( 1) 1y= − − +
Standard form for sideways
parabola: x = a(y-k)2
+h
a = -1 (opens left), vertex (h, k): (1, 1)
Same size and shape as x=y2
3
2
0
[(2 ) ( )]y y y dy− − −∫
9
2
=](https://image.slidesharecdn.com/calc7-1b-120130214107-phpapp01/85/Calc-7-1b-4-320.jpg)
Calc 7.1b
- 1. 7.1b Area of a Region7.1b Area of a Region Between Two CurvesBetween Two Curves Integration as an Accumulation Process
- 2. So far this chapter we have used a representative element – in this case a representative rectangle - to adapt integration to the area between curves. In future sections we will continue to do the same thing. We will come up with a representative element, then using precalculus and geometry formulas, adapt integration to find an accumulation of those elements. [ ( ) ( )] b a f x g x dx−∫ For example, in this section we used the idea that Area = (height)(width) and we found the area of a representative rectangle to show that ∆Area = [f(x) - g(x)]Δx. Then we summed up all those representative rectangles with integration to get
- 3. Ex 6 p.451 Describing Integration as an Accumulation Process Find the area of the region bounded by the graph of y = 4 – x2 and the x- axis. Describe the integration as an accumulation process. ( ) 2 2 2 [ 4 0]x dx − − −∫The parabola will intersect the x-axis at x = 2 ( ) 2 2 2 4 0x dx − − − =∫ As we let the representative rectangle move from left to right, we can see area accumulating. ( ) 1 2 2 5 4 3 x dx − − − =∫ ( ) 0 2 2 16 4 3 x dx − − =∫ ( ) 1 2 2 27 4 3 x dx − − =∫ ( ) 2 2 2 32 4 3 x dx − − =∫
- 4. Another example. Find area when horizontal rectangle makes sense. ( ) (2 )f y y y= − ( )g y y= − Graph. Find intersections by setting f(y)=g(y). 2y – y2 = -y 3y – y2 = 0 y = 0, y = 3 (2 )x y y= − 2 2y y= − + 2 ( 2 1 1)y y= − − + − 2 (( 1) 1)y= − − − 2 ( 1) 1y= − − + Standard form for sideways parabola: x = a(y-k)2 +h a = -1 (opens left), vertex (h, k): (1, 1) Same size and shape as x=y2 3 2 0 [(2 ) ( )]y y y dy− − −∫ 9 2 =
- 5. As you are creating these integrals, you need to be asking, “is this formula going to work for every representative rectangle as Δy moves from 0 to 3?”
- 6. 7.1b p. 452/ 27-49 odd
- 7. 7.1b p. 452/ 27-49 odd